aa r X i v : . [ m a t h - ph ] O c t Planar Markovian Holonomy Fields
Franck Gabriel
Author address:
UPMC, 4 Place Jussieu, 75005 Paris (France)
Current address : Mathematics Institute, University Of Warwick, Gibbet HillRd, Coventry CV4 7AL, (United-Kingdom)
E-mail address : [email protected] ontents Introduction viiL´evy processes and planar Markovian holonomy fields viiiBraids ixLayout of the article x
Part 1. Basic Notions N planar graph 23Chapter 3. Planar Markovian Holonomy Fields 253.1. Definitions 253.2. Restriction and extension of the structure group 31Chapter 4. Weak Constructibility 35Chapter 5. Group of Reduced Loops 395.1. Definition and facts 395.2. The example of RL ( N ) 415.3. Family of generators of RL v ( G ) 425.4. Random holonomy fields and the group of reduced loops 47 Part 2. Construction of Planar Markovian Holonomy Fields iiiv CONTENTS
Part 3. Characterization of Planar Markovian Holonomy Fields n -punctured disk 718.2. A de-Finetti theorem for the braid group 728.3. Degeneracy of the mixture 748.4. Processes and the braid group 81Chapter 9. Characterization of Stochastically Continuous in Law WeakDiscrete Planar Markovian Holonomy Fields 859.1. Proof of Theorem 9.1 859.2. Consequences of Theorem 9.1 91Chapter 10. Classification of Stochastically Continuous Strong PlanarMarkovian Holonomy Fields 95 Part 4. Markovian Holonomy Fields G -constraints 9911.2. Splitting of a surface 10011.3. Markovian holonomy fields 10011.4. Partition functions for oriented surfaces 10211.5. Uniform measure and Yang-Mills fields 10311.6. Conjecture and main theorem 105Chapter 12. The Free Boundary Condition on The Plane 10712.1. Free boundary condition on a surface 10712.2. Free boundary condition on the plane 10812.3. Building a bridge between general and planar Markovian holonomyfields 110Chapter 13. Characterization of the Spherical Part of Regular MarkovianHolonomy Fields 115Bibliography 119 bstract This text defines and studies planar Markovian holonomy fields which areprocesses indexed by paths on the plane which takes their values in a compactLie group. These processes behave well under the concatenation and orientation-reversing operations on paths. Besides, they satisfy some independence and in-variance by area-preserving homeomorphisms properties. A symmetry arises in thestudy of planar Markovian holonomy fields: the invariance by braids. For finiteand infinite random sequences the notion of invariance by braids is defined andwe prove a new version of the de-Finetti’s theorem. This allows us to construct afamily of planar Markovian holonomy fields called the planar Yang-Mills fields. Weprove that any regular planar Markovian holonomy field is a planar Yang-Mills field.Planar Yang-Mills fields can be partitioned into three categories according to theirdegree of symmetry: we study some equivalent conditions in order to classify them.Finally, we recall the notion of (non planar) Markovian holonomy fields defined byThierry L´evy. Using the results previously proved, we compute the spherical partof any regular Markovian holonomy field.
Received by the editor 10/24/2016.2010
Mathematics Subject Classification.
Key words and phrases. random field, random holonomy, Yang-Mills measure, L´evy processon compact Lie groups, braid group, de Finetti’s theorem, planar graphs, continuous limit.Supported by the “Contrats Doctoraux du minist`ere fran¸cais de la recherche” and the ERCgrant, “Behaviour near criticality, held by M. Hairer. v ntroduction Yang-Mills theory is a theory of random connections on a principal bundle, thelaw of which satisfies some local symmetry: the gauge symmetry. It was introducedin the work of Yang and Mills, in 1954, in [
YM54 ]. Since then, mathematicianshave tried to formulate a proper quantum Yang-Mills theory. The construction ona four dimensional manifold for any compact Lie group is still a challenge: we willfocus in this article on the 2-dimensional quantum Yang-Mills theory. On a formallevel, a Yang-Mills measure is a measure on the space of connections which lookslike: e − S YM ( A ) DA, where S YM ( A ) is the Yang-Mills action of the connection A , which is the L normof the curvature, and DA is a translation invariant measure on the space of connec-tions. Yet, many problems arise with this formulation, the main of which is thatthe space of connections can not be endowed with a translation invariant measure.It took some time to understand which space could be endowed by a well-definedmeasure.One possibility to handle this difficulty in a probabilistic way is to considerholonomies of the random connections along some finite set of paths: thus, af-ter the works of Gross [ Gro85 ], [
Gro88 ], Driver [
Dri89 ], [
Dri91 ] and Sengupta[
Sen92 ], [
Sen97 ] who constructed the Yang-Mills field for a small class of pathsbut for any surface, it was well understood that the Yang-Mills measure was aprocess indexed by some nice paths. Their construction uses the fact that the holo-nomy process under the Yang-Mills measure should satisfy a stochastic differentialequation driven by a Brownian white-noise curvature. The Yang-Mills measure hasto be constructed on the multiplicative functions from the set of paths to a Liegroup, that is the set of functions which have a good behavior under concatenationand orientation-inversion of paths. This idea was already present in the precur-sory work of Albeverio, Høegh-Krohn and Holden ([
AHKH86a ], [
AHKH88a ],[
AHKH88b ], [
AHKH86b ]).In [
L´ev00 ], [
L´ev03 ] and [
L´ev10 ], L´evy gave a new construction. This con-struction allowed him to consider any compact Lie groups, any surfaces and anyrectifiable paths. Besides, it allowed him to generalize the definition of Yang-Millsmeasure to the setting where, in some sense, the curvature of the random connec-tion is a conditioned L´evy noise. The idea was to establish the rigorous discreteconstruction, as proposed by E. Witten in [
Wit41 ] and [
Wit92 ] and to show thatone could take a continuous limit.The discrete construction was defined by considering a perturbation of a uni-form measure, the Ashtekar-Lewandowski measure, by a density. The continuouslimit was established using the general Theorem 3 . . L´ev10 ]. This theorem viiiii INTRODUCTION must be understood as a two-dimensional Kolmogorov’s continuity theorem andone should consider it as one of the most important theorem in the theory of two-dimensional holonomy fields. In the article [
CDG16 ], G. C´ebron, A. Dahlqvistand the author show how to use this theorem in order to construct generalizationsof the master field constructed in [
AS12 ] and [
L´ev12 ].In the seminal book [
L´ev10 ], L´evy defined also Markovian holonomy fields.This is the axiomatic point of view on Yang-Mills measures, seen as families ofmeasures, indexed by surfaces which have a good behavior under chirurgical oper-ations on surfaces and are invariant under area-preserving homeomorphisms. Theimportance of this notion is that Yang-Mills measures are Markovian holonomyfields. It is still unknown if any regular Markovian holonomy field is a Yang-Millsmeasure but this work is a first step in order to prove so.The axiomatic formulation of the Markovian holonomy fields allows us to un-derstand L´evy processes as one-dimensional planar Markovian holonomy fields.
L´evy processes and planar Markovian holonomy fields
Let G be a compact Lie group. If dim ( G ) ≥
1, we endow the group G witha bi-invariant Riemannian distance d G . If G is a finite group, we endow it withthe distance d G ( x, y ) = δ x,y . There exist two notions of L´evy processes dependingon the definitions of the increments: left increments Y t Y − s or right increments Y − s Y t . We will fix the following convention: in this article, a L´evy process on G isa c`adl`ag process with independent and stationary right increments which begins atthe neutral element. In fact one can use a weaker definition and forgot about thec`adl`ag property and define a L´evy process as a continuous in probability family ofrandom variables ( Y t ) t ∈ R + such that for any t > s ≥ • Y − s Y t has same law as Y t − s , • Y − s Y t is independent of σ ( Y u , u < s ), • Y = e a.s.Let Y be a L´evy process on G . Let us denote by D ( R ) the set of integrablesmooth densities on R . For any vol ∈ D ( R ), one can define a measure E vol on G R such that, under E vol , the canonical projection process ( X t ) t ∈ R has the law of (cid:0) Y vol (] −∞ ,t ]) (cid:1) t ∈ R . The family (cid:0) E vol (cid:1) vol ∈D satisfies three properties: -Area-preserving increasing homeomorphism invariance: Let us con-sider ψ , an increasing homeomorphism of R . Let vol and vol ′ be twosmooth densities in D ( R ). Let us suppose that ψ sends vol on vol ′ . Themapping ψ induces a measurable mapping from G R to itself which we willdenote also by ψ and which is defined by: ψ (( x t ) t ∈ R ) = (cid:0) x ψ ( t ) (cid:1) t ∈ R . It is then easy to see that E vol = E vol ′ ◦ ψ − . For example, for any real t ∈ R and any bounded function f on G : E vol ′ (cid:2) f ( X ψ ( t ) ) (cid:3) = E (cid:2) f ( Y vol ′ (] −∞ ,ψ ( t )]) ) (cid:3) = E (cid:2) f ( Y vol (] −∞ ,t ]) ) (cid:3) = E vol (cid:2) f ( X t ) (cid:3) . -Independence: Let vol be a smooth density in D ( R ). Let [ s , t ] and[ s , t ] be two disjoint intervals. Under E vol , σ (cid:0) ( X − s X t ) , s ≤ s < t ≤ t (cid:1) is independent of σ (cid:0) ( X − s X t ) , s ≤ s < t ≤ t (cid:1) . RAIDS ix -Locality property:
Let vol and vol ′ be two smooth densities in D ( R ). Let t be a real such that vol | ] −∞ ,t ] = vol ′| ] −∞ ,t ] . The law of ( X t ) t ≤ t is thesame under E vol as under E vol ′ .Let us consider a family of measures ( E vol ) vol ∈D ( R ) on G R ; we say that itis stochastically continuous if, for any vol ∈ D ( R ), for any sequence ( t n ) n ∈ N , if t n converges to t ∈ R ∪ {−∞} , E vol ( d G ( X t n , X t )) −→ n →∞
0, where we recall that( X t ) t ∈ R is the canonical projection process and where, by convention, X −∞ is theconstant function equal to the neutral element e . If ( E vol ) vol ∈D ( R ) is stochasticallycontinuous and satisfies the three axioms stated above then there exists a L´evyprocess ( Y t ) t ∈ R + such that, for any smooth density vol in D ( R ), the canonicalprojection process ( X t ) t ∈ R has the law of (cid:0) Y vol (] −∞ ,t ]) (cid:1) t ∈ R .With these axioms in mind, looking in Section 3.1 at the definitions of planarMarkovian holonomy fields, the reader can understand why we can consider L´evyprocesses as one-dimensional planar Markovian holonomy fields. The surprisingfact that we will prove in this paper is that the family of regular two-dimensionalplanar Markovian holonomy fields is not bigger than the set of one-dimensionalplanar Markovian holonomy fields. Braids
The most innovative idea of this paper is to introduce for the very first timethe braid group in the study of two-dimensional Yang-Mills theory. This is also oneof the main ingredient in the article [
CDG16 ].The braid group is an object which possesses different facets: a combinatorial, ageometric and an algebraic one. One can introduce the braid group using geometricbraids: this construction allows us to have a graphical and combinatorial frameworkto work with. Since it is the most intuitive construction, we quickly present it sothat the reader will be familiar with these objects.
Proposition . For any n ≥ , let the conguration space C n ( R ) of n in-distinguishable points in the plane be (cid:0) ( R ) n \ ∆ (cid:1) / S n where ∆ is the union of thehyperplanes { x ∈ ( R ) n , x i = x j } . The fundamental group of the configurationspace C n ( R ) is the braid group with n strands B n : B n = π (cid:0) C n ( R ) (cid:1) . Every continuous loop γ in C n ( R ) parametrized by [0 ,
1] and based at thepoint (cid:0) (1 , , ..., ( n, (cid:1) can be seen as n continuous functions γ j ∈ C (cid:0) [0 , , R (cid:1) suchthat, if we set σ : j γ j (1) for any j ∈ { , , n } , the following conditions hold:1- ∀ j ∈ { , ..., n } , γ j (0) = ( j, , σ ∈ S n , ∀ t ∈ [0 , , ∀ j = j ′ , γ j ( t ) = γ j ′ ( t ) . The function γ j is given by the image of γ by the projection π j : (cid:0) R (cid:1) n → R . Wecall γ a geometric braid since if we draw the ( γ j ) nj =1 in R , we obtain a physicalbraid. One can look at Figure 1 to have an illustration of this fact.With this point of view, the composition of two braids is just obtained by gluingtwo geometric braids, taking then the equivalence class by isotopy of the new braidas shown in Figure 2. In this paper, we will take the convention that, in order tocompute β β , one has to put the braid β above the braid β . INTRODUCTION
Figure 1.
A physical braid β . = Figure 2.
The multiplication of two braids.
Figure 3.
A two dimensional diagram representation of β .As we see in Figure 3, one can represent a braid by a two dimensional diagram(or, to be correct, classes of equivalence of two-dimensional diagrams) that we call n -diagrams. This representation can remind the reader the representation of anypermutation by a diagram, yet, in this representation of braids, one rememberswhich string is above an other at each crossing. It is a well-known result that any n -diagram represents a unique braid with n -strands.Thus, in order to construct a braid, we only have to construct a n -diagram.Besides, every computation can be done with the n -diagrams.For any i ∈ { , , n − } , let β i be the equivalence class of ( γ ij ) nj =1 defined by: ∀ k ∈ { , , n } \ { i, i + 1 } , ∀ t ∈ [0 , , γ ik ( t ) = ( k, , ∀ t ∈ [0 , , γ ii ( t ) = (cid:18) i + 12 (cid:19) − e iπt , ∀ t ∈ [0 , , γ ii +1 ( t ) = (cid:18) i + 12 (cid:19) + 12 e iπt , with the usual convention R ≃ C . As any braid can be obtained by braiding twoadjacent strands, the family ( β i ) n − i =1 generates B n . Layout of the article
Since the theory of Markovian holonomy fields is a newborn theory which mixesgeometry, representation, probabilities, we recall all the tools we need and try to
AYOUT OF THE ARTICLE xi i i+1
Figure 4.
The elementary braid β i .make this paper accessible to any people from any domain of mathematics. Thispaper is in the same time an introduction and a sequel to [ L´ev10 ]. The readershouldn’t be surprised that we copy some of the definitions of [
L´ev10 ] as anyreformulation wouldn’t have been as good as L´evy’s formulation.In Section 1, we recall the classical notions: paths, multiplicative functions,Besides, we supply a lack in [
L´ev10 ]: we decided to develop the notion of randomholonomy fields, as it might be possible, in the future, that some general randomholonomy fields of interest would not be Markovian holonomy fields. Thus, anyproposition in [
L´ev10 ] that could be applied to random holonomy fields is statedin this setting. We study the projection of random holonomy fields on the setof gauge-invariant random holonomy fields and, in the gauge-invariant setting, weexplain how to restrict and extend the structure group. At last, we develop theloop paradigm which, in particular, implies the new Proposition 1.40.The Section 2 is devoted to the theory of planar graphs and the notion of G − G ′ piecewise diffeomorphisms. One of the main results is Corollary 2.31 which statesthat any generic planar graph can be seen, via such diffeomorphism, as a sub-graphof the N -graph.Using the previous sections, we define in Section 3, four different notions of pla-nar Markovian holonomy fields. Under some regularity condition, it will be provedin the paper that the four notions are essentially equivalent. These objects are pro-cesses, indexed by paths drawn on the plane, which are gauge-invariant, invariantunder area-preserving homeomorphisms, which satisfy a weak independence prop-erty and a locality property. We consider the questions of restriction and extensionof the structure group for planar Markovian holonomy fields.The equivalence between the notions of weak discrete and weak continuousplanar Markovian holonomy field is then proved in Section 4, using a theorem ofMoser and Dacorogna.In Section 5 we define the group of reduced group, as L´evy did in [ L´ev10 ], andobtain a generalization of L´evy’s work in the planar case. This allows us to exhibitgeneral families of loops which generate the group of reduced loops of any planargraph.Two sections are devoted to the link between braids and probabilities: Sec-tions 6 and 8. In the first one, after explaining an algebraic definition of the braidgroup, we show how the Artin’s theorem can be applied on the group of reducedloops. We define the notion of invariance by braid for finite sequences of randomvariables. Section 8 is devoted to the geometric point of view on braids and to ade-Finetti-Ryll-Nardzewski’s theorem for random infinite sequences which are in-variant under the action of the braid groups. Under an assumption of independenceof the diagonal-conjugacy classes, one can characterize the invariant by diagonal ii INTRODUCTION conjugation braidable sequences which are sequences of i.i.d. random variables. Inthe end of the section, we apply these results to processes.In Sections 7, 9 and 10, the reader can find the main results about planarMarkovian holonomy fields. Section 6, on finite braid-invariant sequences of ran-dom variables, allow us in Section 7 to construct, for any L´evy process which isself-invariant by conjugation, a planar Yang-Mills field associated with it. Thisconstruction differs from all the previous constructions since it uses neither thenotion of uniform or Ashtekar-Lewandowski measure nor the notion of stochasticdifferential equations. This allows us to consider any self-invariant by conjugationL´evy processes, where before, one had to consider L´evy processes with density withrespect to the Haar measure and which were invariant by conjugation by the struc-ture group G . In Section 9 and 10, using the results of Section 8, we prove thatany regular planar Markovian holonomy field is a planar Yang-Mills field. Besides,we show that one can characterize their degree of symmetry according to the lawof the holonomy associated to simple loops.Since any regular planar Markovian holonomy field is a planar Yang-Mills field,is it possible to show that any Markovian holonomy field is a Yang-Mills field? InSection 11, we answer partly to this question. First we recall the notion of Markov-ian holonomy fields. The free boundary condition expectation is constructed andallows us to make a bridge between Markovian holonomy fields and planar Mar-kovian holonomy fields. The results shown previously allows us to prove Theorem11.23: the spherical part of a regular Markovian holonomy field is equal to thespherical part of a Yang-Mills field.In order to get a more accurate idea of the results shown in this article and thedifferent notions defined in it, one can refer to the diagram Page 118. Acknowledgements.
The author would like to gratefully thank A. Dahlqvistand G. C´ebron for the useful discussions, his PhD advisor Pr. T. L´evy for hishelpful comments and his postdoctoral supervisor, Pr. M. Hairer, for giving himthe time to finalize this article. Also, he wishes to acknowledge the help providedby A. Bouthier who suggested using Jordan’s theorem (Theorem 8.8), and by P.and M.-F. Gabriel for proof-reading the English.The first version of this work has been made during the PhD of the authorat the university Paris 6 UPMC. This final version of the paper was completedduring his postdoctoral position at the University of Warwick where the author issupported by the ERC grant, Behaviour near criticality, held by Pr. M. Hairer. art 1
Basic Notions
HAPTER 1
Backgrounds: Paths, Random MultiplicativeFunctions on Paths
Let M be either a smooth compact surface (possibly with boundary) or theplane R . A measure of area on M is a smooth non-vanishing density on M ,that is, a Borel measure which has a smooth positive density with respect to theLebesgue measure in any coordinate chart. It will often be denoted by vol . We call( M, vol ) a measured surface . We endow M with a Riemannian metric γ and wewill denote by γ the standard Riemannian metric on R . The notion of paths that we will use in this paper is given in the followingdefinition.
Definition . A parametrized path on M is a continuous curve c : [0 , → M which is either constant or Lipschitz continuous with speed bounded below bya positive constant.Two parametrized paths can give the same drawing on M but with differentspeed and we will only consider equivalence classes of paths. Definition . Two parametrized paths on M are equivalent if they differby an increasing bi-Lipschitz homeomorphism of [0 , M is denoted by P ( M ).Actually, the notion of path does not depend on γ since the distances definedby two different Riemannian metric are equivalent. Two parametrized paths pp and pp which represent the same path p share the same endpoints. It is thuspossible to define the endpoints of p as the endpoints of any representative of p . If p is a path, by p (resp. p ) we denote the starting point (resp. the arrival point)of p . From now on, we will not make any difference between a path p and anyparametrized path pp ∈ p . Definition . A path is simple either if it is injective on [0 ,
1] or if it isinjective on [0 ,
1[ and p = p .Later, we will need the following subset of paths. Definition . We define Aff γ ( M ) to be the set of paths on M which arepiecewise geodesic paths with respect to γ .The set of paths Aff γ ( R ) will be simply denoted by Aff ( R ). An other set ofpaths will be very important for our study: the set of loops. Definition . A loop l is a path such that l = l . A smooth loop is a loopwhose image is an oriented smooth 1-dimensional submanifold of M . The set ofloops is denoted by L ( M ). Let m be a point of M . A loop l is based at m if l = m .The set of loops based at m is denoted by L m ( M ).We can define the inverse and concatenation operations on paths. Let p and p be paths, let pp and pp be representatives of these paths and let us supposethat p = p . The inverse of p , denoted by p − , is the equivalence class of theparametrized path t pp (1 − t ) . The concatenation of p and p denoted by p p is the equivalence class of the parametrized path: pp .pp : t (cid:26) pp (2 t ) if t ≤ / ,pp (2 t −
1) if t > / . Definition . A set of paths P is connected if any couple of endpoints ofelements of P can be joined by a concatenation of elements of P .Using concatenation we can introduce a relation on the set of loops. Definition . Let P be a set of paths. Two loops l and l ′ are elementarilyequivalent in P if there exist three paths, a, b, c ∈ P such that { l, l ′ } = { ab, acc − b } .We say that l and l ′ are equivalent in P if there exists a finite sequence l = l , ..., l n = l ′ such that l i is elementarily equivalent to l i +1 for any i ∈ { , ..., n − } . We willwrite it l ≃ P l ′ . Definition . A lasso is a loop l such that one can find a simple loop m ,the meander, and a path s , the spoke, such that l = sms − .A loop has a well-defined origin and orientation. A cycle is a loop in whichone forgets about the endpoint. In a non-oriented cycle, the endpoint and theorientation are forgotten. Definition . We say that two loops l and l are related if and only ifthey can be decomposed as: l = cd , l = dc , with c and d two paths. The setof equivalence classes for the relation defined on L ( M ) is the set of cycles. Theoperation of inversion is compatible with this equivalence. A non-oriented cycleis a pair { l, l − } where l is a cycle. Besides, a cycle is simple if any loop whichrepresents it is simple and it is said smooth if any loop which represents it is smooth.We need a notion of convergence of paths in order to define the continuity ofrandom holonomy fields. The definition makes use of the Riemannian metric γ ,yet the notion of convergence with fixed endpoints will not depend on the choiceof the Riemannian metric. We denote by d γ the distance on M which is associatedwith γ . Definition . Let p and p be two paths of M . Let ℓ ( p ) (resp. ℓ ( p )) bethe length of the path p (resp. p ). We define the distance between p and p as: d l ( p , p ) = inf pp ∈ p ,pp ∈ p sup t ∈ [0 , [ d γ ( pp ( t ) , pp ( t ))] + | ℓ ( p ) − ℓ ( p ) | . The topology induced by d l does not depend on the choice of γ .Let ( p n ) n ≥ be a sequence of paths on M . Let p be a path on M . The sequence( p n ) n ≥ converges to p with fixed endpoints if and only if: • d l ( p n , p ) → n → + ∞ , .2. MEASURES ON THE SET OF MULTIPLICATIVE FUNCTIONS 5 • ∀ n ≥ p n = p and p n = p .We will see that the convergence with fixed endpoints behaves well when oneconsiders images of paths by bi-Lipschitz homeomorphisms. Let us consider ψ alocally bi-Lipschitz homeomorphism from R to itself. Lemma . Let p be a path on the plane, the image of p , ψ ( p ) , is also a path. Proof.
We only have to prove that ψ ( p ) has a finite length. This is a conse-quence of the fact that if c : [0 , → R is a continuous function, the length of c isgiven by sup { t ,...,t n | t <... Let us consider ( p n ) n ∈ N and p which satisfy the conditions of thelemma. L´evy proved in Lemma 1 . . 17 of [ L´ev10 ], that p n converges to p uni-formly when the paths are parametrized at constant speed. Let us denote by( p n ( t )) t ∈ [0 , and ( p ( t )) t ∈ [0 , these parametrized paths. Since ψ is locally Lipschitz,( ψ ( p n ( t ))) t ∈ [0 , converges uniformly to ( ψ ( p ( t ))) t ∈ [0 , when n goes to infinity andthere exists a real R such that the speed of ψ ( p n ) for any integer n and the speedof ψ ( p ) are bounded by R . An application of Lemma 1 . . 18 of [ L´ev10 ] and thetriangular inequality allows us to assert that the length of ψ ( p n ) converges to thelength of ψ ( p ). This proves that d l ( ψ ( p n ) , ψ ( p )) converges to zero when n goes toinfinity. (cid:3) Using this notion of convergence, one can define a notion of density. Thefollowing lemma was proved by L´evy in Proposition 1 . . 12 of [ L´ev10 ]. Lemma . The set of paths Aff γ ( M ) is dense in P ( M ) for the convergencewith fixed endpoints. One has to be careful when working with the convergence with fixed endpoints.For example, the set of paths whose images are concatenation of horizontal andvertical segments is not dense in P ( R ). Indeed, one condition in order to have theconvergence with fixed endpoints is that the length of the paths converges to thelength of the limit path. But, for any path p which can be written as a concatenationof horizontal and vertical segments, ℓ ( p ) ≥ || p − p || , where || . || is the usual L norm on R , yet this inequality does not hold for a general path p . In this section, the presentation differs from the one of [ L´ev10 ]: new definitionsand new results already appear in this section.From now on, except if specified, G is a compact Lie group, with the usualconvention that a compact Lie group of dimension 0 is a finite group. The neutralelement will be denoted by e . We endow G with a bi-invariant Riemannian distance d G . If G is a finite group, we endow it with the distance d G ( x, y ) = δ x,y . We denoteby M ( G ) the space of finite Borel positive measures on G . 1. BACKGROUND Let P be a subset of P ( M ) and let L be a set of loopsin P . Definition . A function h from P to G is multiplicative if and only if: • h ( c − ) = h ( c ) − for any path c in P such that c − ∈ P , • h ( c c ) = h ( c ) h ( c ) for any paths c and c in P which can be concate-nated and such that c c ∈ P .We denote by M ult ( P, G ) the set of multiplicative functions from P to G . Afunction from L to G is pre-multiplicative over P if and only if: • it is multiplicative, • for any l and l ′ in L which are equivalent in P , we have: h ( l ) = h ( l ′ ).We denote by M ult P ( L, G ) the set of pre-multiplicative functions over P .We will often make the following slight abuse of notation. Notation . Let c be a path in P . If a multiplicative function h is notspecified in a formula, h ( c ) will stand for the function on M ult ( P, G ): h ( c ) : M ult ( P, G ) → Gh h ( c ) . The notion of equivalence of loops, as stated in Definition 1.7, is important dueto the following remark. Remark . Let h be in M ult ( P, G ) and let l, l ′ be loops in P . A simpleinduction and the multiplicative property of h imply that if l ≃ P l ′ then h ( l ) = h ( l ′ ).Let P be a set of paths and let Q be a freely generating subset of P in thesense that: • any path in P is a finite concatenation of elements of Q , • no element of Q can be written as a non-trivial finite concatenation ofpaths in Q ∪ Q − , • Q ∩ Q − = ∅ .Then we have the identification: M ult ( P, G ) ≃ G Q . (1.1)This is the edge paradigm for multiplicative functions. The novelty of the approachwe have in this paper is to put the emphasis on the loop paradigm for gauge-invariant random holonomy fields. The first paradigm is interesting for generalrandom holonomy fields on surfaces, yet the second seems to be more appropriatefor gauge-invariant random holonomy fields on the plane. Remark . All the following definitions and propositions deal with multi-plicative functions on a set of paths P . All of them extend to G T , with T ⊂ R .Indeed, if P = ∪ r ∈ T { c r , c − r } , with c r being the path on the plane based at 0and going clockwise once around the circle of center (0 , r ) and radius r , then M ult ( P, G ) ≃ G T . We will now endow the space of multiplicative functions with a σ -field in orderto be able to speak about measures on M ult ( P, G ). Definition . The Borel σ -field B on M ult ( P, G ) is the smallest σ -fieldsuch that for any paths c , ..., c n and any continuous function f : G n → R , themapping h f ( h ( c ) , ..., h ( c n )) is measurable. .2. MEASURES ON THE SET OF MULTIPLICATIVE FUNCTIONS 7 Definition . A random holonomy field µ on the set P is a measure on( M ult ( P, G ) , B ). If P = P ( M ), we call it a random holonomy field on M .Let µ be a random holonomy field on P : the weight of µ is µ ( ). One candefine a regularity notion for random holonomy fields. Definition . A random holonomy field µ on P is stochastically continuousif for any sequence ( p n ) n ≥ of elements of P which converges with fixed endpointsto p ∈ P , Z M ult ( P,G ) d G ( h ( p n ) , h ( p )) µ ( dh ) −→ n →∞ . (1.2)The measure µ is locally stochastically -H¨older continuous if for any compact set S ⊂ M , for any measure of area vol on M , there exists K > l ∈ P bounding a disk D such that l ⊂ S : Z M ult ( P,G ) d G ( e, h ( l )) µ ( dh ) ≤ K p vol ( D ) , (1.3)where e is the neutral element of G .A family of random holonomy fields, ( µ ) µ ∈F , with each µ defined on some set P µ , is uniformly locally stochastically -H¨older continuous if the constant K inequation (1.3) is independent of the random holonomy field in F . Let us review the twomains results on which the construction of random holonomy fields is based. Notation . Let J and K be two subsets of P ( M ) such that J ⊂ K . Therestriction function from M ult ( K, G ) to M ult ( J, G ) will be denoted by ρ J,K . If M ⊂ M ′ are two surfaces, we denote by ρ M,M ′ the restriction function ρ P ( M ) ,P ( M ′ ) .The notation is set such that for any J ⊂ K ⊂ L ⊂ P ( M ), ρ J,K ◦ ρ K,L = ρ J,L . The fact that G is a compact group allows us to construct measures on the setof multiplicative functions by taking projective limits of random holonomy fields onfinite subsets of paths. This behavior is very different from what can be observedfor Gaussian measures on Banach spaces. Indeed, in [ L´ev10 ], Proposition 2.2.3,L´evy proved, when F is a collection of finite subsets of P , the next propositionusing an application of Carath´eodory’s extension theorem. We give a proof basedon the Riesz-Markov’s theorem, proof which shows clearly why we only considercompact groups. Proposition . Let F be a collection of subsets of paths on M . We denoteby P their union. Suppose that, when ordered by the inclusion, F is directed: forany J and J in F , there exists J ∈ F such that J ∪ J ⊂ J . For any J ∈ F , let m J be a probability measure on ( M ult ( J, G ) , B ) . Assume that the probability spaces ( M ult ( J, G ) , B , m J ) endowed with the restriction mappings ρ J,K for J ⊂ K form aprojective system. This means that for any J and J in F such that J ⊂ J , onehas m J = m J ◦ ρ − J ,J . Then there exists a unique probability measure m on ( M ult ( P, G ) , B ) such that forany J ∈ F , m J = m ◦ ρ − J,P . 1. BACKGROUND Proof. We endow G P with the product topology. As an application of Ty-chonoff’s theorem it is a compact space. A consequence of this is that M ult ( P, G ),endowed with the restricted topology, is also a compact space as it is closed in G P .Besides, the σ -field B is the Borel σ -field on M ult ( P, G ). Let us consider A the setof cylinder continuous functions, that is the set of functions f : M ult ( P, G ) → R + of the form: f : h f ( h ( p ) , ..., h ( p n )) , for some n ∈ N , some p , ..., p n ∈ P and some continuous function f : G n → R .The set A is a subalgebra of the algebra C ( M ult ( P, G ) , R ) of real-valued contin-uous functions on M ult ( P, G ). This subalgebra separates the points of M ult ( P, G )and contains a non-zero constant function. Due to the Stone-Weierstrass’s theo-rem, A is dense in C ( M ult ( P, G ) , R ). Any function f in A depends only on a finitenumber of paths, so that there exists some J ∈ F such that f can be seen as acontinuous function on M ult ( J, P ). We define: m ( f ) = m J ( f ) , which does not depend on the chosen J ∈ F thanks to the projectivity and multi-plicative properties.We have defined a positive linear functional m on A , the norm of which isbounded by the total weight of any of the measures ( m J ) J ∈F . Thus m can beextended on C ( M ult ( P, G ) , R ) and an application of the Riesz-Markov’s theoremallows us to consider m as a measure on ( M ult ( P, G ) , B ). This is the projectivelimit of ( m J ) J ∈F . (cid:3) The notion of locally stochastically -H¨older continuity allows us to have anextension theorem from some subsets of paths to their closure, as shown in theproof of Corollary 3.3.2 of [ L´ev10 ]. Theorem . Let µ Aff γ ( M ) be a random holonomy field on Aff γ ( M ) . If it islocally stochastically -H¨older continuous then there exists a unique stochasticallycontinuous random holonomy field µ on M such that: µ Aff γ ( M ) = µ ◦ ρ − Aff γ ( M ) ,P ( M ) . For any subset P of P ( M ), a natural group actson M ult ( P, G ), the gauge group , that we are going to describe. Let us fix a subset P of P ( M ) which will stay fixed until the end of the chapter. Definition . Let V = { x ∈ M, ∃ p ∈ P, x = p or x = p } be the set ofendpoints of P . We define the partial gauge group associated with P by setting J P = G V . If P = P ( M ), this group is called the gauge group of M . The group J P acts by gauge transformations on the space M ult ( P, G ): if j ∈ J P , the actionof j on h ∈ M ult ( P, G ) is given by: ∀ c ∈ P, ( j • h )( c ) = j − c h ( c ) j c . Let Q = { c , ..., c n } be a finite set of paths on M . Looking only at the evaluationon c i for i ∈ { , ..., n } , we have the inclusion: M ult ( Q, G ) ⊂ G n . The gauge actionof J Q on M ult ( Q, G ) extends naturally to an action on G n by ( j • g ) i = j − c i g i j c i for any i ∈ { , ..., n } . .2. MEASURES ON THE SET OF MULTIPLICATIVE FUNCTIONS 9 Remark . If l , ..., l n are loops based at a point m , the partial gauge groupis nothing but G and the corresponding action on G n is the diagonal conjugation: j • ( g , ..., g n ) = ( j − g j, ..., j − g n j ) . We denote by (cid:2) ( g , ..., g n ) (cid:3) the equivalence class of ( g , ..., g n ) in G n under thediagonal conjugation action.We now define a sub- σ -field of B , the invariant σ -field. Definition . On M ult ( P, G ), the invariant σ -field, denoted by I , is thesmallest σ -field such that for any paths c , ..., c n in P and any continuous function f : G n → R invariant under the action of J { c ,...,c n } on G n defined after Definition1.24, the mapping h f ( h ( c ) , ..., h ( c n )) is measurable.Let us remark that if M is the disjoint union of two smooth compact surfaces, M = M ⊔ M then M ult ( P ( M ) , G ) ≃ M ult ( P ( M ) , G ) × M ult ( P ( M ) , G ). Be-sides, let I (respectively I , I ) be the invariant σ -field on M ult ( P ( M ) , G ) (re-spectively M ult ( P ( M ) , G ), M ult ( P ( M ) , G )). We have I ≃ I ⊗ I . Locally bi-Lipschitz homeomorphisms between surfaces give rise to some exam-ples of functions which are measurable with respect to the Borel and the invariant σ -fields. Given M and M ′ two smooth compact surfaces, suppose that we are givena locally bi-Lipschitz homeomorphism ψ from M to M ′ , we can construct, for any h in M ult ( P ( M ′ ) , G ), a natural multiplicative function ψ ∗ h on M :( ψ ∗ h ) ( p ) = h ( ψ ( p )) , ∀ p ∈ P ( M ) . This defines a function ψ ∗ : M ult ( P ( M ′ ) , G ) → M ult ( P ( M ) , G ) . The function ψ ∗ is measurable for the Borel and the invariant σ -fields. From now on, we denotealso by ψ the application ψ ∗ .On the invariant σ -field on M ult ( P, G ), any measure is of course invariantby the gauge transformations. Explicitly, for any measure µ on ( M ult ( P, G ) , I ),for any measurable continuous function f from ( M ult ( P, G ) , I ) to R and for any j ∈ J P : Z M ult ( P,G ) f ( j • h ) dµ ( h ) = Z M ult ( P,G ) f ( h ) dµ ( h ) . (1.4)The following definition is less trivial as the following class of gauge-invariant mea-sures is not equal to the collection of all measures. Definition . Let µ be a random holonomy field on P . We say that µ isinvariant under gauge transformations if and only if the Equality (1.4) holds forany continuous function f from ( M ult ( P, G ) , B ) to R and any j ∈ J P . Remark . Let µ be a gauge-invariant random holonomy field on P . Let p a path in P which is not a loop: p = p . Then under µµ ( ) , h ( p ) has the law of aHaar random variable. Indeed, applying the gauge transformation which is equalto 1 everywhere except at p or p , where its value is set to be an arbitrary elementof G , we see that the law of h ( p ) is invariant by left- and right-multiplication.There exists a one-to-one correspondence between measures on ( M ult ( P, G ) , I )and gauge-invariant measures on ( M ult ( P, G ) , B ). The next proposition is similarto the results of [ Bae94 ]. For any positive integer n , for any continuous function f on G n and any set of paths { c , ..., c n } in P , we define the function ˆ f J c ,...,cn suchthat, for any g , ..., g n in G :ˆ f J c ,...,cn ( g , ..., g n ) = Z J c ,...,cn f ( j • ( g , ..., g n )) dj, (1.5)where dj is the Haar measure on J c ,...,c n . Notation . If µ is a finite measure on a measurable space (Ω , A ) and if B ⊂ A is a sub- σ -field, by µ |B , we denote the image of µ by the identity map:(Ω , A ) → (Ω , B ). Proposition . For any measure µ on ( M ult ( P, G ) , I ) , there exists aunique gauge-invariant random holonomy field on P which will be denoted eitherby ˆ µ or µ b , such that ˆ µ |I = µ. Proof. The uniqueness of ˆ µ follows from the upcoming Proposition 1.37. Letus prove its existence. We will define ˆ µ by the fact that for any measurable function f : G n → R + and any n -tuple c , ..., c n of elements of P :ˆ µ ( f ( h ( c ) , ..., h ( c n ))) = µ (cid:16) ˆ f J c ,...,cn ( h ( c ) , ..., h ( c n )) (cid:17) . Let us consider a finite set of paths in P , P = { c , ..., c n } . Let us considerthe natural inclusion ι : M ult ( P , G ) ⊂ G n given by the evaluations on c , ..., c n .The equalities ˆ µ P ( f ) = µ (cid:16) ˆ f J c ,...,cn ( h ( c ) , ..., h ( c n )) (cid:17) for any continuous functionon G n define a linear positive functional on C ( G n ). By compactness of G n , ap-plying the theorem of Riesz-Markov, it gives a measure ˆ µ P on G n , the support ofwhich is easily seen to be a subset of ι ( M ult ( P , G )). We can thus look at theinduced measure on M ult ( P , G ) denoted by ˆ µ |M ult ( P ,G ) . The family of measures (cid:0) ˆ µ |M ult ( P ,G ) (cid:1) P ⊂ P, P < ∞ forms a projective family of measures for the inclusionof sets. Thus, by Proposition 1.22, it defines a measure on ( M ult ( P, G ) , B ). (cid:3) Let us introduce a notion which will be important in the definition of planarmarkovian holonomy fields. Let µ be a random holonomy field on P . Definition . Let P and P be two families of paths in P . We will saythat ( h ( p )) p ∈ P and ( h ( p )) p ∈ P are I -independent if and only if, for any finitefamily ( p i ) ni =1 in P , any finite family ( p i ) mi =1 in P and any continuous function f : G n → R (resp. g : G m → R ) invariant under the action of J { p ,...,p n } (resp. J { p ,...,p m } ), the following equality holds: µ h f (cid:0) ( h ( p i )) ni =1 (cid:1) g (cid:16) ( h ( p j )) mj =1 (cid:17)i = µ (cid:2) f (cid:0) ( h ( p i )) ni =1 (cid:1)(cid:3) µ h g (cid:16) ( h ( p j )) mj =1 (cid:17)i . (1.6)This is equivalent to say that under µ , the two σ -fields σ ( h ( p ) : p ∈ P ) ∩ I and σ ( h ( p ) : p ∈ P ) ∩ I are independent.Let us remark that the invariant σ -field on G which we denote by I (2) is, ingeneral, different from the product I ⊗ I where I is the invariant σ -field of G .When G = S , this fact is implied by the following assertion:11 = (cid:8) [( σ, σ ′ )] , ( σ, σ ′ ) ∈ S (cid:9) = ( { [ σ ] , σ ∈ S } ) = 9 . In particular, if ( X, Y ) is a random vector such that X is I -independent of Y , theknowledge of the laws of the random conjugacy classes [ X ] and [ Y ] does not allowus to reconstruct the law of the random diagonal conjugacy class [( X, Y )]. In the .2. MEASURES ON THE SET OF MULTIPLICATIVE FUNCTIONS 11 following remark, we will see that in some very special cases, the I -independenceis equivalent to the independence. Remark . Let P and P be two sets of paths such that their sets of end-points V P and V P are disjoint. The two families ( h ( p )) p ∈ P and ( h ( p )) p ∈ P , definedon ( M ult ( P, G ) , B , µ ), are I -independent if and only if they are independent.We only have to prove that the I -independence implies the independence. Letus suppose that they are I -independent. If f and g are real-valued continuousfunctions on G n and G m respectively, we denote by f ⊗ g the function from G n × G m to R defined by: f ⊗ g ( x , ..., x n , x n +1 , ..., x n + m ) = f ( x , ..., x n ) g ( x n +1 , ..., x n + m ) . With this notation and the notation (1.5), since the two families P and P havedisjoint sets of endpoints, \ ( f ⊗ g ) J P ∪ P = b f J P ⊗ b g J P , where the partial gauge group was defined in Definition 1.24. Thus, using thegauge-invariance of µ , µ [ f (( h ( p )) p ∈ P ) g (( h ( p )) p ∈ P )] = µ [( f ⊗ g ) (( h ( p )) p ∈ P , ( h ( p )) p ∈ P )]= µ h \ ( f ⊗ g ) J P ∪ P (( h ( p )) p ∈ P , ( h ( p )) p ∈ P ) i = µ h b f J P ⊗ b g J P (( h ( p )) p ∈ P , ( h ( p )) p ∈ P ) i = µ h b f J P (( h ( p )) p ∈ P ) i µ (cid:2)b g J P (( h ( p )) p ∈ P ) (cid:3) = µ [ f (( h ( p )) p ∈ P )] µ [ g (( h ( p )) p ∈ P )] . This proves that the two families ( h ( p )) p ∈ P and ( h ( p )) p ∈ P are independent.Let us introduce the main ingredient in order to construct gauge-invariantrandom holonomy fields: the loop paradigm for multiplicative functions. From nowon, P will be connected, stable by concatenation and inversion, m is an endpointof P and we recall that L m is the set of loops in P based at m . Lemma . The loop paradigm for the multiplicative functions is: M ult ( P, G ) /J P ≃ M ult P ( L m , G ) /J L m . (1.7) Proof. There exists a natural restriction function: r : M ult ( P, G ) /J P → M ult P ( L m , G ) /J L m . Let us show that there exists an application ι : M ult P ( L m , G ) /J L m → M ult ( P, G ) /J P , such that r ◦ ι = id and ι ◦ r = id .The proof uses the ideas used in order to prove Lemma 2.1.5 of [ L´ev10 ]. Forany endpoint v of P , let q v be a path in P joining m to v . This is possible since wesupposed that P was connected. We set q m to be the trivial path. Then, for anypath p in P we define l ( p ) = q p pq − p . One can look at the Figure 1 to have a betterunderstanding of l ( p ). For any h in M ult P ( L m , G ), we define for any path p , ι ( h )( p ) = h ( l ( p )) . pq p_ q p_ Figure 1. Construction of the loop l ( p ).This is a multiplicative function. Let us show, for example, that it is compatiblewith the concatenation operation. For any h ∈ M ult P ( L m , G ) and any paths p and p ′ in P such that p = p ′ , the following sequence of equalities holds: ι ( h )( pp ′ ) = h ( l ( pp ′ )) = h ( q p pp ′ q − p ′ ) = h ( q p pq − p q p ′ p ′ q − p ′ )= h ( q p ′ p ′ q − p ′ ) h ( q p pq − p ) = h ( l ( p ′ )) h ( l ( p )) = ι ( h )( p ′ ) ι ( h )( p ) , where in the third equality we used the fact that h is an element of M ult P ( L m , G )and not only in M ult ( L m , G ).Thus ι is an application from M ult P ( L m , G ) to M ult ( P, G ). This applica-tion ι defines a function, that we will also call ι from M ult P ( L m , G ) /J L m to M ult ( P, G ) /J P . Indeed, if j ∈ J L m ≃ G , h ∈ M ult P ( L m , G ) and p ∈ P : ι ( j • h )( p ) = j • h ( l ( p )) = j ( m ) − h ( l ( p )) j ( m ) = ˜ j • ι ( h )( p ) , where ˜ j is the constant function which is equal to j . Let us show that ι ◦ r = id :for any h ∈ M ult ( P, G ),( ι ( r ( h ))( p )) p ∈ P = r ( h )( l ( p )) = (cid:16) h ( q p pq − p ) (cid:17) p ∈ P = (cid:16) h ( q p ) − h ( p ) h ( q p ) (cid:17) p ∈ P , thus, in M ult ( P, G ) /J P , we have the equality ( ι ( r ( h ))( p )) p ∈ P = ( h ( p )) p ∈ P . Theequality r ◦ ι = id is even easier. (cid:3) From the proof of Lemma 1.33, one also gets the following lemma. Lemma . There exists an application: ι : M ult P ( L m , G ) → M ult ( P, G ) which is measurable for the Borel σ -field and such that, for any loop l ∈ L m , thefollowing diagram is commutative: M ult P ( L m , G ) ι / / h ( l ) & & ▼▼▼▼▼▼▼▼▼▼▼ M ult ( P, G ) h ( l ) y y ssssssssss G An other consequence of Lemma 1.33 is Lemma 2 . . L´ev10 ] given below. .2. MEASURES ON THE SET OF MULTIPLICATIVE FUNCTIONS 13 Lemma . For any paths c , ..., c n in P and any measurable function f : G n → R invariant under the action of J c ,...,c n on G n , there exist n loops l , ..., l n in P based at m and a measurable function ˜ f : G n → R invariant under the diagonalaction of G such that: f ( h ( c ) , ..., h ( c n )) = ˜ f ( h ( l ) , ..., h ( l n )) . Lemma 1.35 allows us to reduce the family of variables that I has to makemeasurable: we only have to look at finite collections of loops based at the samepoint. This leads us to the Definition 2 . . L´ev10 ], which, in our case, is aresult and not a definition. Proposition . The invariant σ -field I on M ult ( P, G ) is the smallest σ -field such that for any positive integer n , any loops l , ..., l n based at m and anycontinuous function f : G n → R invariant under the diagonal action of G , themapping h f ( h ( l ) , ..., h ( l n )) is measurable. Another consequence of Lemma 1.35 is the following proposition. Proposition . If µ and ν are two stochastically continuous gauge-invariantrandom holonomy fields on P , the two following assertions are equivalent: (1) µ and ν are equal, (2) there exist an endpoint m of P and A m a dense subset of L m for theconvergence with fixed endpoints, such that for any integer n , any n -tupleof loops l , ..., l n in A m and any continuous function f : G n → R invariantunder the diagonal action of G , Z M ult ( P,G ) f ( h ( l ) , ..., h ( l n )) dµ ( h ) = Z M ult ( P,G ) f ( h ( l ) , ..., h ( l n )) dν ( h ) . If the random holonomy fields are not stochastically continuous, the propositionstill holds if one replaces A m by L m . Remark . The first consequence of this proposition is the change of basepoint invariance property of gauge-invariant random holonomy fields. For the sakeof simplicity, let us consider µ a gauge-invariant random holonomy field on M . Letus consider a bijection ψ : M → M and let us consider for any point x of M , p x apath from ψ ( x ) to x . Then the random holonomy field which has the law of:( h ( p x )) x ∈ M • [ h ( p )] p ∈ P under µ , is still gauge-invariant and the last proposition shows that µ and the newrandom holonomy field are equal. Thus, for any paths p , ...p n , we have the equalityin law: (cid:16)h ( h ( p x )) x ∈ M • [ h ( p )] p ∈ P i ( p i ) (cid:17) ni =1 = ( h ( p i )) ni =1 under µ . For example, if l , ..., l n are n loops based at m and if s is a path from m ′ to m , under a gauge-invariant measure µ , (cid:0) h ( sl s − ) , ..., h ( sl n s − ) (cid:1) has the samelaw as ( h ( l ) , ..., h ( l n )). In this section, for the sake of simplicity, we will suppose that M is connected.However, all results could be easily extended to the non-connected case. Thanksto Lemma 1.33 and Proposition 1.30, constructing a gauge-invariant random holo-nomy field µ becomes easier. We recall that P is a connected set of paths, stableby concatenation and inversion. Proposition . Let m be an endpoint of P . Suppose that for any finitesubset L of loops in P based at m , we are given a gauge-invariant measure µ L on M ult P ( L, G ) such that, when endowed with the natural restriction functions, (( M ult P ( L, G ) , B ) , µ L ) is a projective family. Then there exists a unique gauge-invariant random holonomy field µ on P such that for any finite subset L of loopsin P based at m , one has: µ L = µ ◦ ρ − L,P . Proof. The uniqueness of such a measure comes from a direct application ofProposition 1.37.Let us prove the existence of the measure µ . Let L m be the set of loops in P based at m . Using a slight modification of Proposition 1.22, we can consider theprojective limit µ L m of ( µ L f ) L f ⊂ L m , L f < ∞ , defined on ( M ult P ( L m , G ) , B ) andwhich is gauge-invariant. The set P satisfies the assumptions of Lemma 1.34: letus consider a measurable application ι from M ult P ( L m , G ) to M ult ( P, G ) givenby this lemma. We define the measure: µ = (cid:0) ( µ L m ◦ ι − ) |I (cid:1) b , where we remind the reader that ( ) b is the notation for the extension of mea-sures from the invariant σ -field to the Borel σ -field given by Proposition 1.30. Bydefinition, it is defined on the Borel σ -field on M ult ( P, G ) and it is gauge-invariant.If L is a finite subset of loops in P based at m , thanks to the definitions of ι and µ L m , ( µ L m ◦ ι − ) ◦ ρ − L,P = µ L . The gauge-invariance of µ L implies that (cid:16) ( µ L ) |I (cid:17) b = µ L . This leads us to the conclusion: µ L = µ ◦ ρ − L,P . (cid:3) In particular, if we combine this proposition with Theorem 1.23, we get thefollowing result. Proposition . Let γ be a Riemannian metric on M , let m be a point of M .Suppose that for any finite subset L of loops in Aff γ ( M ) based at m , we are given agauge-invariant measure µ L on M ult P ( L, G ) such that (( M ult P ( L, G ) , B ) , µ L ) is aprojective family of uniformly locally stochastically -H¨older continuous random ho-lonomy fields. Then there exists a unique stochastically continuous gauge-invariantrandom holonomy field µ on M such that for any finite subset L of loops in P ( M ) based at m , one has: µ L = µ ◦ ρ − L,P ( M ) . Let H be aclosed subgroup of G . There exists a natural injection i P : ( M ult ( P, H ) , B ) → ( M ult ( P, G ) , B ) . Thus, we can always push forward any H -valued random holo-nomy field by i P in order to define a G -valued random holonomy field. Of course,if a G -valued random holonomy field on P , say µ , is such that there exists a closedgroup H ⊂ G such that for any path p ∈ P , one has h ( p ) ∈ H µ a.s., then we can .2. MEASURES ON THE SET OF MULTIPLICATIVE FUNCTIONS 15 restrict the group to H : for any finite P f ⊂ P it defines a measure on M ult ( P f , H )and we can take the projective limit thanks to Proposition 1 . µ is a H -valued gauge-invariant random holonomy field, µ ◦ i − P is not in general a G -valuedgauge-invariant random holonomy field. The simplest counterexample is to consider P to be reduced to a single loop: a G -valued random variable can be H -invariantbut not G -invariant by conjugation. Thus, in order to extend the structure groupfrom H to G of a H -gauge-invariant random holonomy field µ , one has to consider: (cid:0) ( µ ◦ i − P ) |I (cid:1) b (1.8)the gauge-invariant extension (see Proposition 1.30) to B of the restriction on theinvariant σ -field I of µ ◦ i − P . Thus, the natural injection is replaced by the followingmap: µ (cid:0) ( µ ◦ i − P ) |I (cid:1) b . Notation . In the following, we will denote µ ◦ ˆ i − P := (cid:0) ( µ ◦ i − P ) |I (cid:1) b .Now, let us consider the problem of restricting a gauge-invariant random ho-lonomy field µ . Thanks to Lemma 1.35, we know that the only important objectsare loops based at m . Hence the question: what can be done with a G -valuedrandom holonomy field such that for any loop or for any simple loop l ∈ L m , µ a.s. h ( l ) ∈ H ? An important remark is that it does not imply that for any path p ∈ P , µ a.s. h ( p ) ∈ H . Indeed, as we have seen in Remark 1.28, for any p such that p = p ,under µ/µ ( ), h ( p ) has the law of a Haar random variable on G . Nevertheless, thefollowing result is true. Proposition . Let µ be a G -valued gauge-invariant random holonomyfield such that for any loop l ∈ L m , h ( l ) ∈ H , µ a.s. Then there exists an H -valuedgauge-invariant random holonomy field µ H such that: µ = µ H ◦ ˆ i − P . Let M be a smooth connected compact surface and let us suppose that P is P ( M ) and that µ is stochastically continuous. The result is still true if for any lasso l based at m , h ( l ) ∈ H , µ a.s. Remark . An important remark is that µ H is not unique. Besides, usingthe group of reduced loops (Section 2.4 of [ L´ev10 ] and the forthcoming Section5.3), one can show in the last case that it is enough that h ( l ) ∈ H , µ a.s., for anysimple loop l based at m . This is due to the fact that for any graph, there exists afamily of generators of the group of reduced loops which can be approximated, forthe convergence with fixed endpoints, by simple loops.We give below the loop-erasure lemma, taken from Proposition 1.4.9 in [ L´ev10 ],that we will use in the proof of Proposition 1.42. Lemma . Let ( M, γ ) be a Riemannian compact surface and let c be a loopin Aff γ ( M ) . There exists in Aff γ ( M ) a finite sequence of lassos l ,..., l p and a simpleloop d with the same endpoints as c such that: c ≃ l ...l p d. Proof of Proposition 1.42. Let us prove the second case, when P = P ( M )and µ is stochastically continuous. The first assertion is easier and can be provedusing the second part of the proof.Let us suppose that for any simple lasso l ∈ L m ( M ), h ( l ) ∈ H , µ a.s. Letus consider γ a Riemannian metric on M . As a consequence of Lemma 1.44, forany loop l ∈ Aff γ ( M ) based at m , h ( l ) ∈ H , µ a.s. Thus, by Lemma 1.13, usingthe stochastic continuity of µ and the fact that H is closed, for any l ∈ L m ( M ), h ( l ) ∈ H , µ a.s.By restricting the measure µ , one can define, for any finite subset L f of L m ( M ),a gauge-invariant measure µ L f on M ult P ( M ) ( L f , H ). As a consequence of Propo-sition 1.39, there exists a unique H -valued gauge-invariant random holonomy field µ H on M such that for any finite subset L f of L m ( M ), µ L f = µ H ◦ ρ − L f ,P ( M ) . This H -valued gauge-invariant random holonomy field µ H satisfies the equality: µ = µ H ◦ ˆ i − P ( M ) . (cid:3) HAPTER 2 Graphs The construction of special random fields, the planar Markovian holonomyfields, uses the notion of graphs. The graphs we consider are not only combinatorialones: we insist that the faces are homeomorphic to an open disk of R . Let M bea either a smooth compact surface with boundary or the plane R . Definition . A pre-graph on M is a triple G = ( V , E , F ) such that: • E , the set of edges, is a non-empty finite set of simple paths on M , stableby inversion, such that two edges which are not each other’s inverse meet,if at all, only at some of their endpoints, • V , the set of vertices, is the finite subset of M given by S e ∈ E { e, e } , • F , the set of faces, is the set of the connected components of M \ S e ∈ E e (cid:0) [0 , (cid:1) .Any pre-graph G = ( V , E , F ) whose bounded faces F ∈ F are homeomorphic to anopen disk of R is called a graph on M . Remark . By Proposition 1.3.10 in [ L´ev10 ], if G is a graph on M then ∂M can be represented by a concatenation of edges in E .Due to the last definition, any pre-graph G = ( V , E , F ) is characterized by itsset of edges E . Thus, in order to construct a pre-graph, we will only define its setof edges. We will often use the following graph. Example . Let l be a simple loop on R . We denote by G ( l ) the graph on R composed of l and l − as unique edges. When M is homeomorphic to a sphere, we will consider that ( { m } , ∅ , M \ { m } )is a graph for any m ∈ M . Definition . A graph is connected if and only if any two points of V are theendpoints of the same path in P ( G ). A connected graph on R will be also called a finite planar graph ; its set of faces is composed of one unbounded face denoted by F ∞ and a set F b of bounded faces. Definition . Let G be a graph on M, P ( G ) is the set of paths obtainedby concatenating edges of G . The set of loops in P ( G ) is denoted by L ( G ) and if v ∈ V , L v ( G ) is the set of loops in L ( G ) based at v .For any smooth connected compact surface with boundary M embedded in R ,a graph on M can be considered as a finite planar graph. This kind of graphs, ofinterest later, will be called embedded graphs on R . 178 2. GRAPHS Definition . An embedded graph on R is a graph on a smooth connectedcompact surface with boundary M embedded in R .The two definitions of graphs on R seen here are in fact almost equivalent. Anembedded graph is obviously a graph on R and a direct consequence of Propositions1 . . 24 and 1 . . 26 of [ L´ev10 ] is the following result. Proposition . Every finite planar graph on R is a subgraph of an embeddedgraph. The intersection of a graph G = ( V , E , F ) with a subset A of R is the pre-graph( V ′ , E ′ , F ′ ), denoted by G ∩ A , such that E ′ = { e ∈ E , e ∩ A * { e, e }} . This allowsus to define the notion of a planar graph. For any positive real r , let D (0 , r ) be theclosed ball of center (0 , 0) and radius r in R . Definition . A planar graph G = ( V , E , F ) is a triple of sets which representthe vertices, the edges and the faces which are linked by the same relations asin Definition 2.1 and for which there exists an increasing unbounded sequence ofpositive reals ( r n ) n ∈ N such that for each integer n , G ∩ D (0 , r n ) is a finite planargraph. Example . We consider N as a planar graph, the edges being the verticaland horizontal segments between nearest neighbors. Notation . Sometimes, one wants to consider connected graphs whoseedges are in a given subset A of P ( M ). We denote by G ( A ) the set of connectedgraphs G = { V , E , F } such that E ⊂ A .In the notions of graph exposed above, the edges are non-oriented, which meansthat there is no preference between e and e − for any edge e . Definition . An orientation on a graph G is the data of a subset E + of E such that E + ∩ ( E + ) − = ∅ and E + ∪ ( E + ) − = E . Given an orientation E + on G , for each subset J of E , we denote by J + the set J ∩ E + . In the following we will need to understand the action of orientation-preservinghomeomorphisms on the set of graphs. Definition . Let G and G ′ be two finite planar graphs. They are home-omorphic if there exists an orientation-preserving homeomorphism ψ which sends G on G ′ . We will denote it by ψ ( G ) = G ′ and by definition, this means that ψ induces a bijection S ψ G from the set V of vertices of G to the set V ′ of vertices of G ′ and a bijection E ψ G from the set E of edges of G to the set E ′ of edges of G ′ . Thesebijections are defined by: S ψ G ( v ) = ψ ( v ), for any v ∈ V , E ψ G ( e ) = ψ ( e ), for any e ∈ E . Definition . Let ψ and ψ ′ be two orientation-preserving homeomorphismsof R which send G on G ′ . The homeomorphisms ψ and ψ ′ are equivalent on G ifand only if S ψ G = S ψ ′ G and E ψ G = E ψ ′ G . .2. GRAPHS AND HOMEOMORPHISMS 19 We would like to have an easy way to know if two finite planar graphs arehomeomorphic. For that, an important notion is the cyclic order of the outgoingedges at a vertex. Definition . Let G = ( V , E , F ) be a finite planar graph. Let v be a vertexand let E v be the set of edges e ∈ E such that e = v . For any e ∈ E v , let e p be aparametrized path which represents e . We define: r = min (cid:26)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v − e p (cid:18) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , e ∈ E v (cid:27) . Let r ∈ ]0 , r [. For each e ∈ E v , we define s e ( r ) ∈ (cid:2) , (cid:3) as the first time e p hits theboundary of D (0 , r ): s e ( r ) = inf (cid:26) t ∈ (cid:20) , (cid:21) , || v − e p ( t ) || = r (cid:27) . The cyclic permutation of E v , corresponding to the cyclic order of the points { e p ( s e ( r )) , e ∈ E v } on the circle ∂ D (0 , r ) oriented anti-clockwise, does not dependon the chosen r ∈ ]0 , r [: it is the cyclic order of the edges at the vertex v denotedby σ v .A consequence of Jordan-Sch¨onfliess theorem is the Heffter-Edmonds-Ringelrotation principle, stated in Theorem 3 . . MT01 ]. Theorem . Let G = ( V , E , F ) and G ′ = ( V ′ , E ′ , F ′ ) be two finite planargraphs such that the following assertions hold: (1) there exists a bijection S : V → V ′ , (2) there exists a bijection E : E → E ′ such that for any e ∈ E , E (cid:0) e − (cid:1) = E ( e ) − , (3) for any edge e ∈ E , S (cid:0) e (cid:1) = E ( e ) , (4) for any vertex v ∈ V , σ S ( v ) = E ◦ σ v ◦ E − .Then there exists an orientation-preserving homeomorphism ψ : R → R such that ψ ( G ) = G ′ and ψ induces the two bijections S and E . If one considers only piecewise affine edges, the theorem can be applied topre-graph with affine edges.Later we will need the notion of diffeomorphisms at infinity. The motivation willappear in Lemma 12.10 where we show that the free boundary condition expectationon the plane associated with a Markovian holonomy field is a planar Markovianholonomy field. In the following definition, D (0 , R ) c is the complement set of theclosed disk centered at 0 and of radius R . Definition . A homeomorphism ψ of R is a diffeomorphism at infinityif there exists a real R such that ψ | D (0 ,R ) c is a diffeomorphism.Each time we consider a homeomorphism from a close domain delimited by aJordan curve to an other domain delimited by an other Jordan curve, we can extendit as a diffeomorphism at infinity. Indeed, using the Carath´eodory’s theorem forJordan curves, we can suppose that both domains are the unit disk. In this case,the result follows from the following lemma. Lemma . Let D be the closed disk of center and radius . Let ψ : ∂ D → ∂ D be a homeomorphism. There exists a diffeomorphism Ψ : D c → D c such that forany x ∈ ∂ D , lim y → x Ψ( y ) = ψ ( x ) . Besides, if ψ preserves the orientation, Ψ will also preserve the orientation. Proof. Let η be a smooth even positive function supported on [ − , r > η r ( . ) = ( r − − η (( r − − . ). The family( η r ) r> is a smooth even approximation to the identity when r goes to 1.There is a natural bijection Φ between the set of homeomorphisms of ∂ D andthe set Hom R ∂ D of strictly increasing or decreasing continuous functions f from R to R such that f − Id is 1-periodic. Let ψ : ∂ D → ∂ D be a homeomorphism of thecircle. We define the smooth function Ψ by:Ψ : D c → D c re iπθ re iπ (Φ( ψ ) ∗ η r )( θ ) . Since ψ is continuous on the disk, the function Φ( ψ ) is uniformly continuous.Thus Φ( ψ ) ∗ η r converges uniformly to Φ( ψ ) as r tends to 1. This implies that forany x ∈ ∂ D , lim y → x Ψ( y ) = ψ ( x ). Besides, for any real r > 1, the convolution with η r sends Hom R ∂ D on itself: this implies that Ψ is bijective. Since Ψ is differentiable, itremains to show that the Jacobian of Ψ is strictly positive. Yet, for any x ∈ D c , onlythe module of x is involved in the calculation of the module of Ψ( x ): the Jacobianmatrix is triangular. Since η r is even for any r > ψ ) is strictly increasing(or decreasing), the derivative of Φ( ψ ) ∗ η r is strictly positive (or negative). Thesetwo facts imply that the Jacobian matrix of Ψ is invertible, thus the function Ψ isa diffeomorphism. The last assertion about the orientation-preserving property isstraightforward. (cid:3) The graphs with piecewise affine edges are interesting when one considers aspecial partial order on graphs studied in [ L´ev10 ]. Definition . Let G and G ′ be two planar graphs. We say that G ′ is finerthan G if P ( G ) ⊂ P ( G ′ ). We denote it by G G ′ .In fact, in Lemma 1.4.6 of [ L´ev10 ], L´evy showed that this partial order is notdirected. Yet, one can, by restricting it to a dense subspace of graphs, make itdirected: for this, the edges of the graphs which we consider must be in a goodsubspace as defined below. Definition . Let P be a subset of P ( M ). A good subspace A of P is adense subset of P for the convergence with fixed endpoints such that for any finitesubset { c , ..., c n } of A there exists a graph G such that { c , ..., c n } ⊂ P ( G ).If A is a good subspace, G ( A ) endowed with is directed. The following lemmais a reformulation of Proposition 1 . . L´ev10 ]. Lemma . For any Riemannian metric γ on M , the set Aff γ ( M ) is a goodsubspace for P ( M ) . .4. GRAPHS AND PIECEWISE DIFFEOMORPHISMS 21 There are other natural examples of good subspaces of P ( M ). For example,Baez in [ Bae94 ] used the good subspace of piecewise real-analytic paths in P ( R ) inorder to define the Ashtekar and Lewandowski uniform measure. Another exampleof good subspace is used in the articles [ Sen92 ] and [ Sen97 ].By definition, any path in M can be approximated by a sequence of paths in A if A is a good subspace. But Aff γ ( M ) satisfies the stronger property which roughlyasserts that G ( Aff γ ( M )) is “dense” for a certain notion in the set of planar graphs.The next theorem is a direct consequence of Proposition 1 . . . in [ L´ev10 ]. It hasto be noticed that, in the proof of Proposition 1 . . . in [ L´ev10 ], the measure ofarea does not have to be the measure of area associated with the chosen Riemannianmetric. For the next theorem, let us suppose that M is an oriented compact surfacewith boundary. Theorem . Let G = ( V , E , F ) be a graph on M . Let γ be a Riemannianmetric on M and let vol be a measure of area on M . There exists a sequence offinite planar graphs (cid:0) G n = ( V n , E n , F n ) (cid:1) n ∈ N in G (cid:0) Aff γ ( M ) (cid:1) such that: (1) for any integer n , there exists ψ n an orientation-preserving homeomor-phism of M such that ψ n ( G ) = G n . (2) V n = V , (3) for any edge e ∈ E , ψ n ( e ) converges to e for the convergence with fixedendpoints, (4) for any face F ∈ F , vol ( ψ n ( F )) −→ n →∞ vol ( F ) . Another interesting property of G ( Aff ( R )) is the fact that any generic finiteplanar graph with piecewise affine edges can be sent by a piecewise smooth appli-cation on a subgraph of the N planar graph. We will prove this in Section 2.5, butbefore, we need to gather a few facts about graphs and triangulations. Definition . Let G be a finite planar graph in G (cid:0) Aff ( R ) (cid:1) . It is simple ifthe boundary of any face of G is a simple loop. It is a triangulation if any boundedface is a non degenerate triangle. Definition . Let G be a finite planar graph in G (cid:0) Aff ( R ) (cid:1) . A mesh of G is a simple graph G ′ in G (cid:0) Aff ( R ) (cid:1) such that G G ′ . A triangulation of G is atriangulation T such that G T and the unbounded face of T is the unboundedface of G .Two triangulations are homeomorphic if they are homeomorphic as finite planargraphs. Definition . Let G and G ′ be two finite planar graphs in G (cid:0) Aff ( R ) (cid:1) . Ahomeomorphism φ : R → R is a G − G ′ piecewise diffeomorphism if the threefollowing assertions hold:(1) φ ( G ) = G ′ ,(2) there exists a mesh G of G (resp G ′ of G ′ ) such that φ ( G ) = G ′ andfor any bounded face F of G , φ | F : F → φ ( F ) is a diffeomorphismwhose Jacobian determinant is bounded below and above by some strictlypositive real numbers and can also be extended on the boundary of F , (3) let F ∞ be the unbounded face of G . The application φ | F ∞ : F ∞ → φ ( F ∞ )is a diffeomorphism.We will say that G is a good mesh for φ .The piecewise diffeomorphisms we will construct will always be of the followingform: they will be the extension (using Lemma 2.17 and the discussion before) ofa piecewise affine homeomorphism from the interior of a piecewise affine Jordancurve to itself. Recall the definition of equivalence defined in Definition 2.13. Proposition . Let G and G be two homeomorphic simple finite planargraphs with piecewise affine edges. Let us choose an orientation-preserving homeo-morphism φ : R → R such that φ ( G ) = G . There exist two triangulations, T of G , T of G and an orientation-preserving G − G piecewise-diffeomorphism ψ such that: (1) T is a good mesh for ψ , (2) ψ and φ are equivalent on G , (3) ψ ( T ) = T .Consequently, the set of orientation-preserving G − G piecewise diffeomorphismsis not empty. In order to prove this proposition, we will need the following result proved inthe paper of Aronov-Seidel-Souvaine ([ ASS93 ]). Theorem . Let Q and Q be two simple n -gons, seen as planar graphswith n vertices. Let us choose an orientation-preserving homeomorphism ψ whichsends Q on Q . Let T (resp. T ) be a triangulation of Q (resp. Q ). Thereexists ˆ T (resp. ˆ T ) a triangulation of Q (resp. Q ), finer than T (resp. T ) andan orientation-preserving homeomorphism ψ ′ such that ψ and ψ ′ are equivalent on Q and ψ ′ ( ˆ T ) = ˆ T . Let G , respectively G , be a simple graph in G (cid:0) Aff ( R ) (cid:1) with only one facedenoted F , respectively F . Let ψ be an orientation-preserving homeomorphismwhich sends G on G . Then there exists a positive integer n such that ∂F and ∂F can be seen as two n -gons such that, when one considers these n -gons as graphs, ψ sends ∂F on ∂F : in order to do so, it is enough to add some vertices on theboundaries of F and F . This remark will allow us to apply Theorem 2.26 to thefaces of simple planar graphs with piecewise affine edges. Let us remark also thatthe homeomorphism ψ between ˆ T and ˆ T in Theorem 2 . 26 can be chosen so thatit is affine on each bounded face of ˆ T . Lemma . Let T and T be two triangulations in the plane. If they arehomeomorphic, there exists a function ψ defined on the union of the bounded facesof T and affine on each bounded face of T such that ψ is an orientation-preservinghomeomorphism which sends T on T . Proof. Let T and T be two homeomorphic triangulations and let φ bean orientation-preserving homeomorphism of R which sends T on T . For anybounded face F of T , we can find an orientation-preserving affine map ψ | F , de-fined on F , such that ψ | F and φ are equivalent on the border ∂F , seen as a graphwith 3 vertices. This map is actually unique.Let us remark that for any triangle T , any x ∈ T and any affine map F , F ( x )depends only on the image by F of the edge which contains x . This allows us toglue the affine maps (cid:0) ψ | F (cid:1) F and to get the desired ψ . (cid:3) .5. THE N PLANAR GRAPH 23 Figure 1. An approximation by a generic graph.We can now prove Proposition 2.25. Proof of Proposition 2.25. Let G and G be two simple homeomorphicfinite planar graphs with piecewise affine edges. Let φ be an orientation-preservinghomeomorphism such that φ ( G ) = G . For any bounded face F of G , F and φ ( F ) are simple polygons. As any polygon can be triangulated, one consequence ofTheorem 2.26 and Lemma 2.27 is that there exists T ,F (resp. T ,F ) a triangulationof F (resp. φ ( F )) and ψ | F a function defined on F , affine on each bounded faceof T ,F , such that ψ | F is an orientation-preserving homeomorphism between T ,F and T ,F and such that ψ | F and φ are equivalent on ∂F . We define T (resp. T )as the triangulation obtained by taking the union of all the triangulations ( T ,F ) F (( T ,F ) F ). As in the proof of Lemma 2.27, we can glue the ψ | F together: thisgives a function ψ | F c ∞ defined on the complementary of the unbounded face F ∞ of G . As G is simple, the boundary of F ∞ is a Jordan curve. Thus, accordingto the discussion we had before Lemma 2.17, we can extend ψ | F c ∞ on F ∞ and theresulting homeomorphism, denoted by ψ , is such that ψ | F ∞ is a diffeomorphism.By construction, ψ is an orientation-preserving G − G ′ piecewise diffeomorphism, φ and ψ are equivalent on G and ψ ( T ) = T . (cid:3) N planar graph We have seen after Lemma 1.13 that the set of piecewise horizontal or verticalpaths is not dense in P ( R ) for the convergence with fixed endpoints. In the follow-ing, we show that, in some sense, we can always inject any graph in the N graphdefined in Exemple 2.9. This property is crucial in the study of planar Markovianholonomy fields. Let G = ( V , E , F ) be a finite planar graph in G (cid:0) Aff (cid:0) R (cid:1)(cid:1) . Weremind the reader that for any v ∈ V , E v is the set of edges e ∈ E such that e = v . Definition . The graph G is generic if for any vertex v ∈ V , E v ≤ Lemma . Let v be a vertex of G . There exists a sequence of generic graphs G n = ( V n , E n , F n ) in G (cid:0) Aff (cid:0) R (cid:1)(cid:1) such that for any n ≥ : (1) v ∈ G n , (2) there exists an injective function L n : L v ( G ) → L v ( G n ) such that for anyloop l ∈ L v ( G ) , L n ( l ) converges with fixed endpoints to l . The notion of generic graphs was defined so that one could send any of suchgraph in the N planar graph. Let us suppose, until the end of the chapter, that G is a generic finite planar graph in G (cid:0) Aff (cid:0) R (cid:1)(cid:1) . Proposition . There exists an orientation-preserving homeomorphism of R , denoted by ψ , such that ψ ( G ) is a subgraph of the N planar graph. Proof. For each v ∈ V , we choose a point ˜ v of N such that the points (˜ v ) v ∈ V are all distinct. For each of these points, we choose a subset E ˜ v of edges in N going out of ˜ v such that E ˜ v = E v . We consider the two pre-graphs:(1) G p with set of edges E p , such that E p = { e p ([0 , ]) , e p represents e ∈ E } , (2) G ′ p such that the set of edges E ′ p is equal to ∪ v ∈ V E ˜ v .Let us define the application S : v ˜ v . Because E ˜ v = E v and thanks to theshape of the graphs, we can choose a bijection E : E p → E ′ p such that the conditionsof Theorem 2 . 15 hold. Using this theorem, there exists an orientation-preservinghomeomorphism ψ such that ψ ( G p ) = G ′ p and ψ induces the two bijections S and E . Let us define G ′ = ψ ( G ): we approximate G ′ in a k N . For k big enoughthis approximation defines a graph ˜ G without new vertices. By construction, theassumptions 1. to 4. of Theorem 2.15 hold for G and ˜ G . Using a dilatation we cansuppose that k = 1. Using Theorem 2.15, there exists an orientation-preservinghomeomorphism ψ which sends G to the subgraph ˜ G of the N planar graph. (cid:3) Corollary . There exists a subgraph G ′ of the N graph such that the setof orientation-preserving G − G ′ piecewise diffeomorphisms is not empty. Proof. Due to Proposition 2.30 there exists a subgraph G ′ of the N planargraph such that G and G ′ are homeomorphic: the graph G ′ is also simple. ByProposition 2.25 the set of orientation-preserving G − G ′ piecewise diffeomorphismsis not empty. (cid:3) HAPTER 3 Planar Markovian Holonomy Fields In this chapter, we define the continuous and discrete planar Markovian holo-nomy fields: these are families of random holonomy fields on subsets of P ( R ) satis-fying an area-preserving homeomorphism invariance and an independence property. First, we define the strong and weak notions of (continuous) planar Markovianholonomy fields. We will use the following notation: if l is a simple loop in R , Int ( l ) will stand for the bounded connected component of R \ l . Definition . A G -valued strong (continuous) planar Markovian holonomyfield is the data, for each measure of area vol on R of a gauge-invariant randomholonomy field E vol on R of weight E vol ( ) = 1, such that the three followingaxioms hold: P : Let vol and vol ′ be two measures of area on R . Let ψ : R → R be a locally bi-Lipschitz homeomorphism which preserves the orientationand which sends vol on vol ′ (i.e. vol ′ = vol ◦ ψ − ). The mapping from M ult ( P ( R ) , G ) to itself induced by ψ , denoted also by ψ , satisfies: E vol ′ ◦ ψ − = E vol . Moreover, let G and G ′ be two finite planar graphs, let φ : R → R bea homeomorphism which preserves the orientation, which sends vol on vol ′ and which sends G on G ′ . The mapping from M ult ( P ( G ′ ) , G ) to M ult (cid:0) P ( G ) , G (cid:1) induced by φ , denoted also by φ , satisfies:( E vol ′ ) |M ult ( P ( G ′ ) ,G ) ◦ φ − = ( E vol ) |M ult ( P ( G ) ,G ) . P : For any measure of area vol on R , for any simple loops l and l suchthat Int ( l ) and Int ( l ) are disjoint, under E vol , the two families: n h ( p ) , p ∈ P (cid:0) Int ( l ) (cid:1)o and n h ( p ) , p ∈ P (cid:0) Int ( l ) (cid:1)o are I -independent. P : For any measures of area on R , vol and vol ′ , if l is a simple loopsuch that vol and vol ′ are equal when restricted to the interior of l , thefollowing equality holds: (cid:0) E vol (cid:1) |M ult ( P ( Int ( l )) ,G ) = (cid:0) E vol ′ (cid:1) |M ult ( P ( Int ( l )) ,G ) . In the study of Markovian holonomy fields, it will be convenient to have thenotion of weak (continuous) planar Markovian holonomy fields. 256 3. PLANAR MARKOVIAN HOLONOMY FIELDS Definition . A G -valued weak (continuous) planar Markovian holonomyfield is the data, for each measure of area vol on R of a gauge-invariant randomholonomy field E vol on Aff (cid:0) R (cid:1) of weight E vol ( ) = 1, such that the three followingaxioms hold: wP : Let vol and vol ′ be two measures of area on R . Let ψ : R → R be a diffeomorphism at infinity which preserves the orientation and whichsends vol on vol ′ (i.e. vol ′ = vol ◦ ψ − ). Let p , ..., p n be paths in Aff (cid:0) R (cid:1) such that for any i ∈ { , ..., n } , p ′ i = ψ ( p i ) is in Aff (cid:0) R (cid:1) . Then for anycontinuous function f : G n → R , E vol h f (cid:0) h ( p ) , ..., h ( p n ) (cid:1)i = E vol ′ h f (cid:0) h ( p ′ ) , ..., h ( p ′ n ) (cid:1)i . wP : For any measure of area vol on R , for any simple loops l and l in Aff (cid:0) R (cid:1) such that Int ( l ) and Int ( l ) are disjoint, under E vol , the twofamilies: n h ( p ) , p ∈ Aff (cid:0) R (cid:1) ∩ P (cid:0) Int ( l ) (cid:1)o and n h ( p ) , p ∈ Aff (cid:0) R (cid:1) ∩ P (cid:0) Int ( l ) (cid:1)o are independent. wP : For any measures of area on R , vol and vol ′ , if l is a simple loopsuch that vol and vol ′ are equal when restricted to the interior of l , thefollowing equality holds: (cid:0) E vol (cid:1) |M ult ( Aff ( Int ( l )) ,G ) = (cid:0) E vol ′ (cid:1) |M ult ( Aff ( Int ( l )) ,G ) . It can seem strange that we replaced the I -independence by the usual indepen-dence in wP , but this was precisely the point of Remark 1.32. As a consequence,any strong planar Markovian holonomy field defines, by restriction, a weak planarMarkovian holonomy field. We will see later that the two notions are equivalentwhen we restrict them to stochastically continuous objects. By G -valued (contin-uous) planar Markovian holonomy fields, we will denote the family of G -valuedstrong or weak (continuous) planar Markovian holonomy fields. Definition . A G -valued planar Markovian holonomy field (cid:0) E vol (cid:1) vol isstochastically continuous if, for any measure of area vol on R , E vol is stochas-tically continuous.A discrete counterpart exists for strong planar Markovian holonomy fields. Definition . A G -valued strong discrete planar Markovian holonomy fieldis the data, for each measure of area vol , for each finite planar graph G , of a gauge-invariant random holonomy field E G vol on P ( G ) of weight E G vol ( ) = 1, such that thefour following axioms hold: DP : Let vol and vol ′ be two measures of area on R , let G and G ′ betwo finite planar graphs. Let ψ be a homeomorphism which preserves theorientation, satisfies ψ ( G ) = G ′ and such that for any F ∈ F b , vol ( F ) = vol ′ ( ψ ( F )). The mapping from M ult ( P ( G ′ ) , G ) to M ult (cid:0) P ( G ) , G (cid:1) in-duced by ψ , denoted also by ψ , satisfies: E G ′ vol ′ ◦ ψ − = E G vol . .1. DEFINITIONS 27 DP : For any measure of area vol on R , for any finite planar graph G , forany simple loops l and l in P ( G ), such that Int ( l ) ∩ Int ( l ) = ∅ , under E G vol , the two families: n h ( p ) , p ∈ P ( G ) ∩ P (cid:0) Int ( l ) (cid:1)o and n h ( p ) , p ∈ P ( G ) ∩ P (cid:0) Int ( l ) (cid:1)o are I -independent. DP : For any measures of area on R , vol and vol ′ , if l is a simple loopsuch that vol and vol ′ are equal when restricted to the interior of l , if G is included in Int ( l ), then the following equality holds: E G vol = E G vol ′ . DP : For any measure of area vol on R , for any finite planar graphs G and G , such that G G : E G vol ◦ ρ − P ( G ) ,P ( G ) = E G vol , where we remind the reader that ρ P ( G ) ,P ( G ) : M ult (cid:0) P ( G ) , G (cid:1) → M ult (cid:0) P ( G ) , G (cid:1) is the restriction map.We will use also the following weak version of discrete planar Markovian holo-nomy fields. Definition . A G -valued weak discrete planar Markovian holonomy fieldis the data, for each measure of area vol , for each finite graph G in G (cid:0) Aff (cid:0) R (cid:1)(cid:1) , ofa gauge-invariant random holonomy field E G vol on P ( G ) of weight E G vol ( ) = 1, suchthat the four following axioms hold: wDP : Let vol and vol ′ be two measures of area on R , let G and G ′ be two simple finite planar graphs in G (cid:0) Aff (cid:0) R (cid:1)(cid:1) . Let ψ be a G − G ′ piecewisediffeomorphism which preserves the orientation. Suppose that for anybounded face F of G , vol ( F ) = vol ′ ( ψ ( F )). Then the mapping from M ult ( P ( G ′ ) , G ) to M ult (cid:0) P ( G ) , G (cid:1) induced by ψ satisfies: E G ′ vol ′ ◦ ψ − = E G vol . wDP : For any measure of area vol on R , for any finite graph G in G (cid:0) Aff (cid:0) R (cid:1)(cid:1) , for any simple loops l and l in P ( G ), such that Int ( l ) and Int ( l ) are disjoint, under E G vol , the two families: n h ( p ) , p ∈ P ( G ) ∩ P (cid:0) Int ( l ) (cid:1)o and n h ( p ) , p ∈ P ( G ) ∩ P (cid:0) Int ( l ) (cid:1)o are independent. wDP : For any measures of area on R , vol and vol ′ , if l is a simple loopsuch that vol and vol ′ are equal when restricted to the interior of l , if G is included in Int ( l ), then the following equality holds: E G vol = E G vol ′ . wDP : For any measure of area vol on R , for any finite planar graphs G and G in G (cid:0) Aff (cid:0) R (cid:1)(cid:1) , such that G G : E G vol ◦ ρ − P ( G ) ,P ( G ) = E G vol , where again ρ P ( G ) ,P ( G ) : M ult (cid:0) P ( G ) , G (cid:1) → M ult (cid:0) P ( G ) , G (cid:1) is therestriction map.Let us remark that the Axioms DP and wDP can be directly deduced re-spectively from DP and wDP by considering the identity function of the plane.Yet, in order to have a similar formulation for continuous and discrete objects wepreferred to keep them in the definitions.As for the continuous objects, any G -valued strong discrete planar Markovianholonomy field defines, by restriction, a weak discrete planar Markovian holonomyfield. By G -valued discrete planar Markovian holonomy fields, we will denote thefamily of G -valued strong or weak discrete planar Markovian holonomy fields. Inany assertion about G -valued discrete planar Markovian holonomy fields, the readerwill have to understand that, in the case we are working with a weak discrete planarMarkovian holonomy field, all the graphs must be in G (cid:0) Aff (cid:0) R (cid:1)(cid:1) . From now on, ifnot specified, all the planar Markovian holonomy fields will be G -valued, thus wewill omit to specify it. Remark . Let (cid:0) E G vol (cid:1) G ,vol be a discrete planar Markovian holonomy field.Using Proposition 1.22, the Axiom DP or wDP allows us to define for anymeasure of area vol and any possibly infinite planar graph G , a unique gauge-invariant random holonomy field E G vol on P ( G ) whose weight E G vol ( ) is equal to 1,such that, for any finite planar graph G f G , E G vol ◦ ρ − P ( G f ) ,P ( G ) = E G f vol . Besides, the family (cid:26)(cid:16) M ult (cid:0) P ( G ) , G (cid:1) , B , E G vol (cid:17) G ∈G ( Aff ( R )) , (cid:0) ρ P ( G ) ,P ( G ′ ) (cid:1) G , G ′ ∈G ( Aff ( R )) , G G ′ (cid:27) is a projective family. There exists a unique gauge-invariant random holonomy fieldon Aff (cid:0) R (cid:1) , whose weight is equal to 1, which we denote by E Aff vol , such that for anyfinite planar graph G f ∈ G (cid:0) Aff ( R ) (cid:1) , E Aff vol ◦ ρ − P ( G f ) , Aff ( R ) = E G f vol . The notions of area-dependent continuity and locally stochastically -H¨oldercontinuity that we are going to define are similar to the notions explained in Defi-nition 3 . . L´ev10 ] that L´evy used for discrete Markovian holonomy fields. Let (cid:0) E G vol (cid:1) G ,vol be a family of random holonomy fields such that for any measure of area vol and any finite planar graph G , E G vol is a random holonomy field on P ( G ). Definition . The family (cid:0) E G vol (cid:1) G ,vol is locally stochastically -H¨older con-tinuous if for any measure of area vol on R , (cid:0) E G vol (cid:1) G is a uniformly locally -H¨oldercontinuous family of random holonomy fields.It is continuously area-dependent if, for any sequence of finite planar graphs G n which are the images of a common graph G by a sequence of area-preservinghomeomorphisms ψ n ( ψ n ( G ) = G n ) and such that vol ( ψ n ( F )) tends to vol ( F ) as n tends to infinity for any bounded face F of G , the following convergence holds: E G n vol ◦ ψ − n −→ n →∞ E G vol , where we denote by ψ n the induced map from M ult (cid:0) P ( G n ) , G (cid:1) to M ult (cid:0) P ( G ) , G (cid:1) .It is regular if it is locally stochastically -H¨older continuous and continuouslyarea-dependent.The new notion of stochastic continuity in law is defined as follow. .1. DEFINITIONS 29 Definition . Let (cid:0) E G vol (cid:1) G ,vol be a family of random holonomy fields suchthat for any measure of area vol and any finite planar graph G , E G vol is a randomholonomy field on P ( G ). The family (cid:0) E G vol (cid:1) G ,vol is stochastically continuous in lawif for any measure of area vol , for any integer m , any finite planar graph G , anysequence of finite planar graphs ( G n ) n ≥ and any sequence of m -tuples of loopsin G n , (( l nk ) mk =1 ) n ∈ N , if there exists a m -tuple of loops in G , ( l k ) mk =1 such that forany i ∈ { , .., k } , l ni converges with fixed endpoints to l i when n goes to infinity,then the law of (cid:0) h ( l nk ) (cid:1) mk =1 under E G n vol converges to the law of (cid:0) h ( l k ) (cid:1) mk =1 under E G vol when n goes to infinity.Let us also remark that the Axioms DP and wDP are not discrete versionsof P and wP since in DP and wDP we do not require that vol ′ is the image of vol by ψ . Thus, it is not obvious that any planar Markovian holonomy field, whenrestricted to graphs, defines a discrete planar Markovian holonomy field. For now,we define the notion of constructibility but later we will show that, under someregularity conditions, any planar Markovian holonomy field is constructible. Definition . Let (cid:0) E vol (cid:1) vol be a weak (resp. strong) planar Markovian ho-lonomy field. It is constructible if the family of measures (cid:16) ( E vol ) |M ult ( P ( G ) ,G ) (cid:17) G ,vol is a weak (resp. strong) discrete planar Markovian holonomy field. Remark . If ( E vol ) vol is a constructible stochastically continuous planarMarkovian holonomy field, its restriction to graphs defines a stochastically contin-uous in law discrete planar Markovian holonomy field.We have seen, in Remark 3.6, that given a strong discrete planar Markov-ian holonomy field (cid:0) E G vol (cid:1) G ,vol , we could define a family of probability measures (cid:0) E Aff vol (cid:1) vol . If (cid:0) E G vol (cid:1) G ,vol is locally stochastically -H¨older continuous, so is E Aff vol forany measure of area vol . By Theorem 1 . 23, one can extend E Aff vol as a stochasticallycontinuous random holonomy field on R , denoted by E vol . We have thus defined (cid:0) E vol (cid:1) vol a family of stochastically continuous random holonomy fields on R .Using a slight modification of Theorem 3 . . L´ev10 ], if (cid:0) E G vol (cid:1) G ,vol is con-tinuously area-dependent then the family (cid:0) E vol (cid:1) vol is a stochastically continuousstrong planar Markovian holonomy field. Let us explain the only difficult part ofthis assertion which is to prove that the axiom P is valid for (cid:0) E vol (cid:1) vol .Using the same arguments as L´evy used in Proposition 3 . . L´ev10 ], if (cid:0) E G vol (cid:1) G ,vol is continuously area-dependent then for any finite planar graph G , forany measure of area vol , E G vol = ( E vol ) |M ult ( P ( G ) ,G ) . Let us remark that it is im-portant, in order to prove this assertion, that we consider all the homeomorphismsin the Axiom DP . As a consequence, (cid:0) E vol (cid:1) vol satisfies the second assertion inAxiom P .It remains to prove the first assertion in Axiom P . Let vol , vol ′ and ψ whichsatisfy the conditions of this first assertion. Let p , ..., p n be paths on the planeand let f be a continuous function on G n . We need to prove that: E vol [ f ( h ( p ) , ..., h ( p n ))] = E vol ′ [ f ( h ( ψ ( p )) , ..., h ( ψ ( p n )))] . (3.1)Let us consider, for any i ∈ { , ..., n } , a sequence of piecewise affine paths ( p ji ) j ∈ N which converges with fixed endpoints to p i when j goes to infinity. Using Lemmas i ∈ { , ..., n } , ψ ( p ji ) converges with fixed endpoints to ψ ( p i )when j goes to infinity. Since E vol is stochastically continuous, it is enough to proveEquation (3.1) when p , ..., p n are piecewise affine paths. But in this case, thereexists a graph G such that { p , ..., p n } ⊂ P ( G ) and ψ ( G ) is also a planar graph:Equality (3.1) is a consequence of the already proven second assertion in Axiom P . In a nutshell, we just proved the following theorem. Theorem . Let (cid:0) E G vol (cid:1) G ,vol be a strong discrete planar Markovian holo-nomy field. If it is regular then there exists a unique stochastically continuous strongplanar Markovian holonomy field (cid:0) E vol (cid:1) vol such that, for any finite planar graph G and any measure of area vol , (cid:0) E G vol (cid:1) G ,vol is the restriction to M ult ( P ( G ) , G ) of E vol : E vol ◦ ρ − P ( G ) ,P ( M ) = E G vol . The unicity is a consequence of Proposition 1.37 and Lemma 1.13. Remark . The proof of L´evy of the axiom A page 123 of [ L´ev10 ] inthe proof of Theorem 3 . . γ ′ is welldefined and is a Riemannian metric. In this article, we corrected this proof byconsidering the modified Axiom P in Definition 3.1 and Axiom A in Definition11.6.The next assertion is a consequence of Theorem 3.11 and Remark 3.10 whenone considers strong discrete planar Markovian holonomy fields. It is a directconsequence of Theorem 1 . 23 and Remark 3.10 for weak discrete planar Markovianholonomy fields. Corollary . Any strong discrete planar Markovian holonomy field whichis regular is stochastically continuous in law.Any weak discrete planar Markovian holonomy field which is locally stochasti-cally -H¨older continuous is stochastically continuous in law. In the rest of the paper, we will mostly work with stochastically continuous inlaw discrete planar Markovian holonomy fields. For any parametrized loop l , for any x in R \ l (cid:0) [0 , (cid:1) , the index of l with respect to x is defined as the integer: n l ( x ) = 12 iπ I l dzz − x . Actually, one needs to approximate uniformly l by piecewise smooth loops and takethe limit. An other way to define the index field is by first constructing the L -functions valued non-random holonomy field on Aff ( R ) which sends l on n l : thiscan be defined as a combinatorial object. Using the L norm on the set of N -valuedfunctions on the plane and considering the Lebesgue measure on the plane, it iseasy to see that this holonomy field is locally stochastically -H¨older continuous.Using Theorem 1.23, we can extend it in order to get a stochastically continuousnon-random planar holonomy field on the plane. This construction shows that n l is square-integrable for any rectifiable loop. This can be obtained also by using theBanchoff-Pohl’s inequality proved in [ Vog81 ]. Since n l takes values in N and sinceany loop is bounded, n l is integrable against any measure of area vol . Besides if l +2 +1- 1 0 Figure 1. The index of a curve.and l are based at the same point, using the additivity of the curve integral, weget: n l l = n l + n l . (3.2)Let D be an element of the Lie algebra g of G . We can now define the index fielddriven by D . Definition . The index field driven by D is the only planar Markovianholonomy field (cid:0) E vol (cid:1) vol such that for any measure of area vol , any loops l , ..., l n based at the same point and any continuous function f from G n to R invariant bydiagonal conjugation, we have: E vol h f (cid:0) h ( l ) , ..., h ( l n ) (cid:1)i = f (cid:16) e D R R n l ( x ) vol ( dx ) , ..., e D R R n ln ( x ) vol ( dx ) (cid:17) . The existence of such a planar Markovian holonomy field is due to the factthat one can consider for any finite family of loops ( l , ..., l n ) based at 0, therandom holonomy field on ( l , ..., l n ) such that ( h ( l ) , ..., h ( l n )) has the law of (cid:16) U e D R R n l ( x ) vol ( dx ) U − , ..., U e D R R n ln ( x ) vol ( dx ) U − (cid:17) , where U is a Haar randomvariable on G . It is a gauge-invariant random holonomy field due to the Equation(3.2) and it is actually a measure on M ult P ( R ) ( { l , ..., l n } , G ). An application ofProposition 1.39 allows us to conclude. An interesting fact with this planar Mar-kovian holonomy field is that it is stochastically continuous and constructible. Itcan also be used in order to add a drift to any holonomy field on the plane. Indeed,if µ is a random holonomy field on the plane, vol be a measure of area and D anelement of the center of g , there exists a planar holonomy field µ D,vol such that forany loops l , ..., l n based at the same point and any continuous function f from G n to R invariant by diagonal conjugation, we have: µ D,vol [ f ( h ( l ) , ..., h ( l n ))] = µ h f (cid:16) e D R R n l ( x ) vol ( dx ) h ( l ) , ..., e D R R n ln ( x ) vol ( dx ) h ( l n ) (cid:17)i . Any regularity which holds for µ holds for µ D,vol . Besides, if ( E vol ) vol is a planarMarkovian holonomy field, so is ( E D,volvol ) vol . We have seen in Section 1.2.5 how to restrict and extend the structure groupof a gauge-invariant random holonomy field: we would like to do the same forplanar Markovian holonomy fields. We will work in the setting of discrete planarMarkovian holonomy fields, but the upcoming Propositions 3.15 and 3.16 can alsobe applied to (continuous) planar Markovian holonomy fields. Let H be a closedsubgroup of G . Let (cid:0) E G vol (cid:1) G ,vol be a H -valued discrete planar Markovianholonomy field. Recall the notation µ ◦ ˆ i − P defined in Notation 1.41. FollowingSection 1.2.5, for any finite planar graph G and any measure of area vol , we can see E G vol as a G -valued gauge-invariant random holonomy field on P ( G ) by considering E G vol ◦ ˆ i − P ( G ) . It is not difficult to verify next proposition. Proposition . The family (cid:0) E G vol ◦ ˆ i − P ( G ) (cid:1) G ,vol is a G -valued discrete planarMarkovian holonomy field. The regularities are the same for the H -valued and the G -valued random holonomy fields. Proposition . Let (cid:0) E G vol (cid:1) G ,vol be a G -valued stochastically continuous inlaw discrete planar Markovian holonomy field. Let us suppose that for any finiteplanar graph G , any vertex v of G , any measure of area vol and any simple loop l ∈ L v ( G ) , h ( l ) ∈ H, E G vol a.s.,then there exists a H -valued stochastically continuous in law discrete planar Mar-kovian holonomy field (cid:0) ˜ E G vol (cid:1) G ,vol such that: E G vol = ˜ E G vol ◦ ˆ i − P ( G ) , for any finite planar graph G and any measure of area vol . The proof of Proposition 3.16 relies heavily on a theorem which will be provedlater, namely Theorem 9.1, thus it will be given page 92. It is more difficult thanone might think to prove this proposition because of the non-unicity of the randomholonomy field µ H in Proposition 1.42. In fact, one can show in general that thenatural choice we made in Proposition 1.42 does not allow one to define a H -valued discrete Markovian holonomy field: we will give an exemple page 92 whichillustrates this fact. Let us explain the problem which appears when one wants torestrict the structure group of a discrete Markovian holonomy field. Let (cid:0) E G vol (cid:1) G ,vol be a G -valued stochastically continuous in law discrete planar Markovian holonomyfield which satisfies the condition of Proposition 3.16. Let G be a finite planar graphand let vol be a measure of area on the plane. It is natural to set:˜ E G vol = \ (cid:16)(cid:0) ( E G vol ) | L v ( G ) ◦ ι − (cid:1) |I H (cid:17) , (3.3)where v is any vertex of G , ι : M ult ( L v ( G ) , H ) → M ult ( P ( G ) , H ) is any mapgiven by Lemma 1.34, I H is the H -invariant σ -field and b is the gauge-invariantextension (where the gauge group is now built on H ) given by Proposition 1.30.Let l and l ′ be two simple loops in P ( G ), with l = v , such that Int ( l ) ∩ Int ( l ′ ) = ∅ as shown in the Figure 2. If the family of measures (cid:0) ˜ E G vol (cid:1) G ,vol just defined abovewas a discrete planar Markovian holonomy field, then h ( l ) and h ( l ′ ) would beindependent under ˜ E G vol . But if p is the path from v to l ′ used to define ι and if f , f are two continuous functions on H invariant by conjugation by H , we have:˜ E G vol (cid:16) f (cid:0) h ( l ) (cid:1) f (cid:0) h ( l ′ ) (cid:1)(cid:17) = E G vol (cid:16) f (cid:0) h ( l ) (cid:1) f (cid:0) h ( pl ′ p − ) (cid:1)(cid:17) . (3.4)In the r.h.s. appear the two loops l and pl ′ p − which are not of null intersection(as they share at least v ) and only appear functions invariant by conjugation by H and not by G . This does not allow us to split the expectation into a product. .2. RESTRICTION AND EXTENSION OF THE STRUCTURE GROUP 33 l lp v' Figure 2. Two simple loops l , l ′ with a path p joining l to l ′ .HAPTER 4 Weak Constructibility In this section, any weak continuous planar Markovian holonomy field is shownto be constructible. Let us state a proposition which is a direct consequence of animportant theorem of Moser and Dacorogna in [ DM90 ]. Let Leb be the Lesbeguemeasure on R . Proposition . Let Q be an open simple n -gon in R . Let f and g be twostrictly positive functions on Q which are in C ( Q ) ∩ C ( Q ) . Suppose that: Z Q f d Leb = Z Q gd Leb . Then there exists φ ∈ Diff ( Q ) ∩ Diff ( Q ) , a homeomorphism of Q which restrictsto a diffeomorphism of Q , such that: g. Leb | Q = (cid:0) f. Leb | Q (cid:1) ◦ φ − . and φ ( x ) = x for any x ∈ ∂Q . Proof. In [ DM90 ], page 15 the authors define for any positive integer k , aproperty ( H k ) for open subsets of R n . They show in Theorem 7 of the same paper,that for any positive integer k , any open domain Ω which satisfies ( H k ), any positivefunctions f and g in C k (Ω) with f + f and g + g bounded and which satisfy: Z Ω f d Leb = Z Ω gd Leb , there exists φ ∈ Diff (Ω) ∩ Diff (Ω) with φ ( x ) = x on ∂ Ω such that: g.d Leb | Ω = (cid:0) f.d Leb | Ω (cid:1) ◦ φ − . Besides, Proposition A. H k ) for every k ≥ 1. The proposition follows from thisdiscussion. (cid:3) Theorem . Any weak planar Markovian holonomy field is constructible. Proof. Let ( E vol ) vol be a weak planar Markovian holonomy field. Let usconsider (cid:0) E G vol (cid:1) G ,vol , the family of random holonomy fields that we get by restricting( E vol ) vol on M ult ( P ( G ) , G ) for any finite planar graph G in G (cid:0) Aff (cid:0) R (cid:1)(cid:1) . Asexplained before Definition 3.9, we only have to check that the Axiom wDP issatisfied by (cid:0) E G vol (cid:1) G ,vol .Let vol and vol ′ be two measures of area on R . Consider G and G ′ twosimple finite planar graphs in G (cid:0) Aff ( R ) (cid:1) . Let ψ be an orientation-preserving G − G ′ piecewise diffeomorphism. Let us suppose that for any bounded face F of G , vol ( F ) = vol ′ ( ψ ( F )). 356 4. WEAK CONSTRUCTIBILITY Let F ′∞ be the unbounded face of G ′ . Let us suppose that we managed toconstruct an orientation-preserving diffeomorphism at infinity Φ on R such that: vol ′| ( F ′∞ ) c = (cid:0) vol ◦ (Φ ◦ ψ ) − (cid:1) | ( F ′∞ ) c , (4.1) Φ | G ′ = Id | G ′ . (4.2)As G ′ is a simple graph, the boundary of F ′∞ is a simple loop. Applying the Axiom wP and using the condition (4.1): E G ′ vol ′ = E G ′ vol ◦ (Φ ◦ ψ ) − . Yet by condition (4.2), G ′ = Φ( G ′ ) = Φ ◦ ψ ( G ). Thus, as an application of Axiom wP , we get: E G ′ vol ′ = E G ′ vol ◦ (Φ ◦ ψ ) − = E Φ ◦ ψ ( G ) vol ◦ (Φ ◦ ψ ) − = E G vol , which is the desired equality.It remains to construct an orientation-preserving diffeomorphism at infinity Φon R satisfying the two conditions (4.1) and (4.2). This will be done by applyingtwice the Proposition 4.1. Let G be a good mesh for ψ . First we regularizethe measure vol ◦ ψ − , which does not have a smooth density, by applying theproposition for each face of the mesh G . Then we transport the resulting measureof area on vol ′ by applying again Proposition 4.1 for each bounded face of G ′ .Let us fix a measure of area vol ′′ such that, for any bounded face F of G , vol ′′ ( ψ ( F )) = vol ( F ). Let us consider any bounded face F of G which is in-cluded in a bounded face of G . By definition of a G − G ′ piecewise diffeomorphismand the definition of a good mesh for ψ , ψ | F is a diffeomorphism from F to ψ ( F ) which are two simple n -gons. Thus, f vol | ψ ( F ) = vol | F ◦ ( ψ | F ) − defines ameasure with strictly positive smooth density on ψ ( F ). Using the condition onthe Jacobian determinant of ψ , this smooth density can be extended as a strictlypositive continuous function on ψ ( F ). Using Proposition 4.1, we can consider φ | ψ ( F ) ∈ Diff ( ψ ( F )) ∩ Diff (cid:16) ψ ( F ) (cid:17) with φ | ψ ( F ) ( x ) = x for any x ∈ ∂ψ ( F ) suchthat: vol ′′| ψ ( F ) = f vol | ψ ( F ) ◦ ( φ | ψ ( F ) ) − . Let us finally set φ | ψ ( F ∞ ) = Id | ψ ( F ∞ ) , where F ∞ is the unbounded face of G . Thanks to the boundary condition on φ | ψ ( F ) for any face F of G , we canglue together all the homeomorphisms φ | ψ ( F ) constructed for each face F of G .It defines an orientation-preserving diffeomorphism at infinity φ on R such that vol ′′| ψ ( F ∞ ) c = ( vol ◦ ( φ ◦ ψ ) − ) | ψ ( F ∞ ) c and ( φ ) | G ′ = Id | G ′ .For any bounded face F of G ′ , we have: vol ′′ ( F ) = vol ( ψ − ( F )) = vol ′ ( F ) . Besides, G ′ is a simple graph: we can apply Proposition 4.1 for any bounded face F of G ′ in order to transport vol ′′| F on vol ′| F . Applying the same arguments (gluing thehomeomorphisms as we just did) allows us to construct an orientation-preservinghomeomorphism φ such that:( vol ′ ) | ( F ′∞ ) c = ( vol ′′ ◦ φ − ) | ( F ′∞ ) c ( φ ) | G ′ = Id | G ′ , . WEAK CONSTRUCTIBILITY 37 where we recall that F ′∞ = ψ ( F ∞ ) is the unbounded face of G ′ . The orientation-preserving diffeomorphism at infinity Φ = φ ◦ φ satisfies the two conditions (4.1)and (4.2). (cid:3) HAPTER 5 Group of Reduced Loops In order to construct planar Markovian holonomy fields, we need to study thegroup of reduced loops. In order to construct a gauge-invariant random holonomyfield on P , it is enough to construct a measure on M ult P ( L, G ) for any set L ofloops of P : this was the loop paradigm explained in Lemma 1.33. Let G be a finiteplanar graph, let v be a vertex of G , let L be the set of loops L v ( G ) and let P beequal to P ( G ), then: M ult P ( L, G ) = Hom ( π ( G , v ) , G ∨ ) , where G ∨ is the group based on the same set at G , endowed with the multiplication . ∨ such that x. ∨ y = yx for any x, y ∈ G and π ( G , v ) is the fundamental group of G based at v . In this chapter we study the group π ( G , v ). Let us fix, until the end on the section, a finite planar graph G = ( V , E , F ) anda vertex v of G . The group of based reduced loops RL v ( G ) is the fundamentalgroup of G based at v : RL v ( G ) = π ( G , v ). For convenience we define it using acombinatorial point of view, as L´evy does in Section 1.3.4 of [ L´ev10 ].Let l be a loop in P ( G ). Recall the definition of equivalence of paths explainedin Definition 1.7. The equivalence class of l in P ( G ), denoted by [ l ] ≃ , containsa unique element of shortest combinatorial length, which is said to be reduced.Besides, if l and l are two loops in P ( G ) based at v , [ l l ] ≃ depends only on[ l ] ≃ and [ l ] ≃ . Thus, it is equivalent to speak about equivalence classes or aboutreduced paths and the set of reduced paths is endowed with an internal operation. Definition . The set of reduced loops in P ( G ) based at v will be denotedby RL v ( G ). Let l and l be two loops in RL v ( G ), we define l × l = [ l l ] ≃ .Endowed with this operation, RL v ( G ) is a group. The existence of the inverseof a loop l based at v is due to the fact that [ ll − ] ≃ = [1 v ] ≃ , where 1 v is the trivialpath constant to v . In the following, we will denote the reduced product of l with l by l l rather than l × l . In the following we will study RL v ( G ) using familiesof lassos. Let us state a simple, yet crucial lemma about lassos. Lemma . Two lassos in L v ( G ) whose meanders represent the same cycle areconjugated in RL v ( G ) . Proof. Let l and l ′ be two lassos based at v . They can be written as l = sms − and l ′ = s ′ m ′ s ′− , where s and s ′ are respectively the spoke of l and l ′ . As theloops m and m ′ are related, there exist c and d two paths such that m = cd and m ′ = dc . Let us denote by p the loop s ′ c − s − , then l ′ = plp − . (cid:3) 390 5. GROUP OF REDUCED LOOPS Definition . A loop in G is called a facial lasso if it is a lasso and itsmeander represents a non-oriented facial cycle of G .The exact definition of facial cycle, an oriented or non-oriented cycle whichrepresents the boundary of a face, is given in [ L´ev10 ], in Definition 1.3.13. Forany face F of G , we will denote by ∂F both the non-oriented and oriented facialcycles associated with F . If we specify that ∂F is oriented, we will consider theanti-clockwise orientation.In the following, we address the problem of creating families of lassos whichgenerate the whole group RL v ( G ). The well-known Lemma 5.6 provides a solutionof this problem which is not adapted to our context, but will nevertheless be thedeparture point of our discussion. In order to state it, we need the definition of aspanning tree. Definition . A spanning tree T of G is a subset of E such that: • if an edge e is in T , e − is also in T , • the set of non degenerate loops in T is empty, • V = (cid:8) e, e ∈ E (cid:9) .If G is composed of a unique edge e which is a loop, the set { e } is considered as aspanning tree of G . A rooted spanning tree is the data of a spanning tree T and avertex v of G .Let T be a spanning tree of G rooted at v . The restriction of T to a subgraph G ′ = ( V ′ , E ′ , F ′ ) of G is defined as following: if E = { l, l − } with l a loop, then T ′ = { l } and in the other cases we consider T ′ = T ∩ E ′ . Definition . Let u and w be two vertices of G . The path [ u, w ] T is theunique injective path in T joining u to w . For any edge e ∈ E , we set l e,T =[ v, e ] T e [ e, v ] T . Lemma . Let E + be an orientation of G . The group RL v ( G ) is freely gen-erated by the loops (cid:8) l e,T : e ∈ ( E \ T ) + (cid:9) . Proof. We only have to prove that (cid:0)(cid:8) l e,T : e ∈ ( E \ T ) + (cid:9) , RL v ( G ) (cid:1) satisfiesthe universal property of free groups: given any function f from (cid:8) l e,T : e ∈ ( E \ T ) + (cid:9) to a group G there exists a homomorphism φ : RL v ( G ) → G such that φ ( l e,T ) = f ( l e,T ) , for any e ∈ ( E \ T ) + .Let G be any group and let 1 be its neutral element. We recall the Equa-tion (1.1), in Subsection 1.2, which shows that one can construct a multiplicativefunction from P ( G ) to G by specifying the value on E + . In the definition of multi-plicative functions, we asked that the function reverses the order of multiplication.Only for this proof, we will suppose that it preserves the order. This means that if g ∈ M ult ( P ( G ) , G ), then for any path p and p in P which can be concatenated, g ( p p ) = g ( p ) g ( p ). Let f be a function from (cid:8) l e,T : e ∈ ( E \ T ) + (cid:9) to G . Wedefine the element φ in G E + by: φ ( e ) = (cid:26) f ( l e,T ) , if e ∈ ( E \ T ) + , , otherwise.This defines an element of M ult ( P ( G ) , G ), called also φ , which restriction on L v ( G )induces a homeomorphism from RL v ( G ) to G . Beside, for any path p in T , φ ( p ) = 1. .2. THE EXAMPLE OF RL ( N ) 41 O ( i , j ) Figure 1. The lasso L i,j .Let e be any element of ( E \ T ) + : φ ( l e,T ) = φ ([ v, e ] T e [ e, v ] T ) = φ ([ v, e ] T ) φ ( e ) φ ([ e, v ] T )) = f ( l e,T ) . The universal property of free groups holds: RL v ( G ) is the free group generatedby (cid:8) l e,T : e ∈ ( E \ T ) + (cid:9) . (cid:3) Remark . The loops l e,T defined above are lassos and since G is a finiteplanar graph, E \ T ) + = E + − T = E + − V + 1 = F b . RL ( N )We define in this section a family of facial lassos in N which will be importantin Section 9. Even if this family can be studied with the help of the upcomingProposition 5.12, we give an elementary proof that it generates RL ( N ). Notation . Let ( i, j ) and ( k, l ) be couples of reals such that i = k or j = l .We denote by ( i, j ) → ( k, l ) the straight line from ( i, j ) to ( k, l ) in R . If j = l and k = i + 1 it will also be denoted by e ri,j ; if i = k and l = j + 1 it will also be denotedby e ui,j . Definition . Let i , j be two non negative integers. Let ∂c i,j be the loopin L ( N ) defined by: ∂c i,j = ( i, j ) → ( i + 1 , j ) → ( i + 1 , j + 1) → ( i, j + 1) → ( i, j )= e ri,j e ui +1 ,j ( e ri,j +1 ) − ( e ui,j ) − . Let p i,j be the path in P ( N ) defined by: p i,j = (0 , → ( i, → ( i, j ) = e r , ...e ri − , e ui, ...e ui,j − . Let L i,j be the reduced loop based at 0: L i,j = (cid:2) p i,j ∂c i,j p − i,j (cid:3) ≃ . One can refer to Figure 1 to have a clear representation of the lasso L i,j . Lemma . The family (cid:0) L i,j (cid:1) ( i,j ) ∈ N is a freely generating subset of RL ( N ) . Figure 2. The spanning tree T . Proof. We only have to work with the finite planar graph: G = N ∩ (cid:8) ( x, y ) , x ≤ k, y ≤ k ′ (cid:9) , where k and k ′ are any positive integers. We remind the reader that the intersectionof a graph with a set was defined before Definition 2.8. Lemma 5.6 implies that RL ( G ) is a free group of rank k × k ′ . Let l be a loop in RL ( G ). We endow thegraph G with the following orientation: from bottom to top, from left to right. Let T be the tree defined by: T = n(cid:0) e ui,j (cid:1) ± , i ∈ { , ..., k } , j ∈ { , ..., k ′ − } o ∪ n(cid:0) e ri, (cid:1) ± , i ∈ { , ..., k − } o . The root of T will be chosen to be (0 , l can be written as the reduced concatenation of some elements of (cid:8) l ± e,T (cid:9) e ∈ ( E \ T ) + , where E is the set of edges of G . Moreover ( E \ T ) + is equal to (cid:8) e ri,j , i ∈ { , ..., k − } , j ∈ { , ..., k ′ } (cid:9) . Since( l e ri,j ,T ) − = L i, L i, ...L i,j − , the family (cid:0) L i,j (cid:1) ( i,j ) ∈ N is a generating subset of RL ( G ) whose cardinal is k × k ′ :it is a freely generating subset of RL ( G ). (cid:3) RL v ( G )In the setting of planar graphs, this section is a generalization of Section 2 . L´ev10 ] about tame generators. The proofs explained here do not use the ideas in[ L´ev10 ] but rather uses a recursive decomposition of graphs. Let G = ( V , E , F ) bea finite planar graph, let v ∈ V and let T be a spanning tree of G rooted at v . Nextdefinition follows Definition 2.4.6. of [ L´ev10 ]. Let ( l e,T ) e ∈ E be the loops defined inDefinition 5.5. Definition . Let F be a bounded face of G and let c F be a simple looprepresenting the facial non-oriented cycle associated with F : it can be written as c F = e ...e n . We define the reduced path l c F ,T = l e ,T ...l e n ,T in RL v ( G ). .3. FAMILY OF GENERATORS OF RL v ( G ) 43 v v c F _ c F c F _ c ,T F [l ] Figure 3. A graph, a spanning tree, a facial cycle: the associatedreduced facial lasso.Let ( c F ) F ∈ F b be such that c F is a representative of the non-oriented facial cycleassociated with F : it is called a family of facial loops of G . We have defined a newfamily of loops (cid:0) l c F ,T (cid:1) F ∈ F b . The difference with Definition 2 . . L´ev10 ] is thatthe choice of c F is not given by the choice of T , there is freedom to choose the basepoint of c F .A remark that we will often use is that, when one changes the root of T from v to an other vertex v ′ , this has the effect to conjugate the family (cid:0) l c F ,T (cid:1) F ∈ F b by[ v ′ , v ] T . This comes from the fact that, for any spanning tree T , any vertices v , v ′ and v ′′ , we have the equality in the set of reduced paths, [ v, v ′′ ] T = [ v, v ′ ] T [ v ′ , v ′′ ] T .Proposition 5.12 and Lemma 5.13 give the two most important properties of thesefamilies of reduced loops. Proposition . For any bounded face F , l c F ,T is a facial lasso based at v whose meander represents the non-oriented facial cycle ∂F . Besides, ( l c F ,T ) F ∈ F b freely generates RL v ( G ) . Proof. The equality l c F ,T = [ v, c F ] T c F [ v, c F ] − T allows us to see that l c F ,T isa lasso of meander c F and spoke [ v, c F ] T .As seen in Lemma 5.6 and Remark 5.7, RL v ( G ) is a free group of rank F b .Thus, it remains to show that ( l c F ,T ) F ∈ F b generates RL v ( G ). Using Lemma 5.6,it is enough to show that for every e ∈ E \ T , l e,T is a product of elements of theform l ± c F ,T . Let e be an edge which is not in T . As T is a tree, there exist c , p and p ′ three simple paths in T which do not intersect, except at the point c = p = p ′ ,such that: • [ v, e ] T = c p, • [ v, e ] T = c p ′ , • the meander m of l e,T is pep ′− .Let v ′ be any point of G inside the meander of l e,T . Since T is a tree, [ v, v ′ ] T mustbegin with the path c . If not, it would create a non degenerate loop in T . Define G ′ (resp. T ′ ) the restriction of G (resp. T ) to the closure of the inside of the meander m of l e,T . An example is given in Figure 4. Let c be the root of T ′ . For everybounded face F of G inside m , l c F ,T = c l c F ,T ′ c − where l c F ,T ′ is the facial lassobased at c defined in G ′ thanks to T ′ .Applying the upcoming Lemma 5.13 to G ′ endowed with T ′ , m can be writtenas a product of lassos of the form l ± c F ,T ′ , thus l e,T can be written as a product oflassos of the form l ± c F ,T . (cid:3) v v Figure 4. The restriction of G used in Proposition 5.12. Lemma . Let us suppose that v is actually a vertex on the boundary of theunbounded face. Let ( F i ) F b i =1 be an enumeration of the bounded faces. Let l ∞ be theonly loop in P ( G ) based at v with anti-clockwise orientation which represents thenon-oriented facial cycle of the unbounded face F ∞ . There exists a permutation σ of { , ..., F b } and an application ǫ : { , ..., n } → {− , } such that the equality l ǫ ( n ) c Fσ ( n ) ,T l ǫ ( n − c Fσ ( n − ,T ...l ǫ (1) c Fσ (1) ,T = l ∞ holds in RL v ( G ) . Besides, for any integer k ∈ { , ..., n } , ǫ ( k ) is equal to if andonly if c F σ ( k ) is oriented anti-clockwise. Proof. In this proof, all the equalities will hold in RL v ( G ): from now onwe will omit to specify this. The last assertion comes from a topological indexargument. Let us suppose that there exists a permutation σ of { , ..., F b } andan application ǫ : { , ..., n } → {− , } such that l ǫ ( n ) c Fσ ( n ) ,T l ǫ ( n − c Fσ ( n − ,T ...l ǫ (1) c Fσ (1) ,T = l ∞ . We can compute the index of l ∞ : n l ∞ = n l cFσ ( n ) ,T ǫ ( n ) + ... + n l cFσ (1) ,T ǫ (1) . For anybounded face of G , namely F , we can evaluate the last equality for any x ∈ F . Thisimplies that for any i ∈ { , ..., n } , 1 = n l cFσ ( i ) ,T ǫ ( i ) , hence the second assertion.Let us show the first part of Lemma 5.13. The proof goes by induction onthe number F b of bounded faces. For a graph with only one bounded face F theresult is true since l ∞ = l ǫc F , with ǫ being − c F . Let us suppose that F b > 1. There exists a unique way to write l ∞ as p e p e ...e n p n with p i a path in T (which can be constant) and e i an edge in E \ T bounding F ∞ for any i in { , ..., n } . Let us decompose the graph G in n subgraphs.The i -th subgraph G i is the part of G which is inside the meander m i of l e i ,T . Thevertex v i = m i will be the chosen point on the boundary of G i , then: • the restriction T i of T to G i is still a spanning tree of G i , • for any bounded face F in G i , l c F ,T = [ v, v i ] T l c F ,T i [ v, v i ] − T , where l c F ,T i is the facial lasso based at v i defined in the graph G i .If n > 1, each of the graphs G i has strictly less than F b bounded faces. Anexample is drawn in Figure 5. By induction the result holds for G i , based at v i andendowed with T i . It follows that l e i ,T , which is equal to [ v, v i ] T m i [ v, v i ] − T , is anordered product of all the facial lasso (or their inverse) associated with the faces F in G i . But the family ( G i ) i induces a partition of the set of bounded faces of G .As l ∞ = l e ,T ...l e n ,T it is now clear that the result holds. .3. FAMILY OF GENERATORS OF RL v ( G ) 45 v v e e Figure 5. Decomposition when n > v v e F v l Figure 6. Decomposition when G l has as many faces as G .It remains the case where n = 1. In this case, l ∞ = pep ′ , with p and p ′ twosimple paths in T and e an edge in E \ T bounding F ∞ . We have to find a new wayof decomposing G in order to apply the induction hypothesis. Let F be the onlybounded face which is surrounded by e . We can suppose c F turning clockwise thusit can be decomposed as c F = ae − b . Consider the loop: l = [ v, c F ] T ae − [ e, v ] T .This is a lasso and as before we consider G l , which is the graph G restricted to theclosure of the interior of the meander m l of l . We base this graph at v l = m l .First of all, if G l has the same number of faces than G , as in Figure 6, then theequality l = l − ∞ must hold and thus one has l ∞ = ([ v, e ] T b [ c F , v ] T ) l − c F . The path˜ l = [ v, e ] T b [ c F , v ] T is a loop based at v which represents the non-oriented facialcycle of the unbounded face of the graph obtained when one removes e to G . Onthis graph, T is still a spanning tree and this graph has one less bounded face. Theinduction hypothesis allows us to conclude.In the case where G l has less faces than G , as in Figure 7, the restriction of T in m l is not a spanning tree. We will define T l to be the restriction of T in Int ( m l ) towhich one adds all the edges in the path a and we set its root equal to v l . With thesemodifications, T l is a spanning tree of G l and for any bounded face F of G l , we have l c F ,T = s l l c F ,T l s − l , where l c F ,T l is the facial lasso based at v l defined in G l thanksto T l and s l is the spoke of l . We define also l ′ = [ v, c F ] T a [ e, v ] T and G l ′ the part of G inside Int ( m l ′ ) of l ′ . The restriction of T to G l ′ is denoted T l ′ . In this case T l ′ is aspanning tree of G l ′ . Besides, for any bounded face F in G l ′ , l c F ,T = s l ′ l c F ,T l ′ s − l ′ ,where l c F ,T l ′ is the facial lasso based at v l ′ = m l ′ defined in G l ′ thanks to T l ′ and s l ′ is the spoke of l ′ . In the case we are studying, G l and G l ′ have strictly less boundedfaces than G , thus we can apply the induction hypothesis. Using the link betweenfacial lassos in G l (resp. in G l ′ ) and in G , there exists an ordering on the bounded e F ba c F v _ e F c F _ c F _ c F _ v Figure 7. Decomposition when G l has less faces than G .faces of G l (resp. G l ′ ) such that l (resp. l ′ ) is the ordered product of the faciallassos ( l ± c F ,T ) F ∈ F bl (resp. ( l ± c F ,T ) F ∈ F bl ′ ), where F bl (resp. F b ′ l ) is the set of boundedfaces of G l (resp. G l ′ ). Since l ∞ = [ v, e ] T e [ e, v ] T = l − l ′ and as any bounded face F of G is either a bounded face of G l or G l ′ , we can conclude that there exists apermutation σ of { , ..., F b } and an application ǫ : { , ..., n } → {− , } such that: l ǫ ( n ) c Fσ ( n ) ,T l ǫ ( n − c Fσ ( n − ,T ...l ǫ (1) c Fσ (1) ,T = l ∞ . This allows us to conclude. (cid:3) Let us finish with a proposition which will be needed in order to prove Propo-sition 7.5. Proposition . Let l and l be two simple loops in G such that Int ( l ) and Int ( l ) are disjoint. There exists a spanning tree T , rooted at v , such that for anyfamily of facial loops ( c F ) F ∈ F b the following assertions hold: (1) for every loop l in P ( G ) included in Int ( l ) , h [ v, l ] T l [ v, l ] − T i ≃ is a productin RL v ( G ) of elements of n l ± c F ,T ; F ∈ F b , F ⊂ Int ( l ) o , (2) for every loop l in P ( G ) included in Int ( l ) , h [ v, l ] T l [ v, l ] − T i ≃ is a productin RL v ( G ) of elements of n l ± c F ,T ; F ∈ F b , F ⊂ Int ( l ) o . Proof. We can decompose l and l as a concatenation of edges of G : l = e ...e n and l = e ...e m . The set (cid:8) e , ..., e n − , e , ..., e m − (cid:9) can be extended as aspanning tree T of the graph G , rooted at v . Thanks to the construction, therestriction T of T to Int ( l ) is a spanning tree of the restriction G of G to Int ( l ). We set v to be equal to e : this is the root of T . Applying Proposi-tion 5.12, for any loop l inside l , h [ v , l ] T l [ v , l ] − T i ≃ is a product of elements of n l ± c F ,T ; F ∈ F b , F ⊂ Int ( l ) o . For any vertex w in G , [ v, w ] T = [ v, v ] T [ v , w ] T in RL v ( G ). Thus for any face F ∈ F b such that F is included in Int ( l ), l c F ,T =[ v, v ] T l c F ,T [ v, v ] − T in RL v ( G ) and for any loop l in P ( G ) included in Int ( l ),[ v, l ] T l [ v , l ] − T is equal in RL v ( G ) to [ v, v ] T [ v , l ] T l [ v , l ] − T [ v, v ] − T . Thus the loop[ v, l ] T l [ v , l ] − T is a product of elements of n l ± c F ,T ; F ∈ F b , F ⊂ Int ( l ) o . The sameholds for the loop l . (cid:3) .4. UNICITY AND CONSTRUCTION 47 Let us explain two applications of the group of reduced loops which concernthe uniqueness and construction of random holonomy fields on the plane. Proposition . Let µ and ν be two stochastically continuous measures on (cid:0) M ult ( P ( R ) , G ) , B (cid:1) which are invariant by gauge transformations. The two as-sertions are equivalent: (1) µ and ν are equal, (2) there exist v ∈ R and A v a good subspace of L v ( R ) , such that for anyfinite planar graph G in G (cid:0) A v (cid:1) which has v as a vertex, there exist a rootedspanning tree T and a family of facial loops ( c F ) F ∈ F b of G such that thelaw of ( h ( l c F ,T )) F ∈ F b is the same under µ and under ν . Proof. It is an easy application of the multiplicative property of random ho-lonomy fields, Proposition 1.37 and Proposition 5.12. (cid:3) Proposition . Suppose that for any finite planar graph G in G (cid:0) Aff ( R ) (cid:1) ,we are given a diagonal conjugation-invariant measure µ G on G F b , a rooted span-ning tree T and a family of facial loops ( c F ) F ∈ F b . For any finite planar graph G , there is only one possibility to extend µ G as a gauge-invariant random field on P ( G ) , which will be also denoted by µ G , such that the law of ( h ( l c F ,T )) F ∈ F b under µ G is the same as the law of the canonical projections under the measure µ G on G F b . If ( µ G ) G ∈G ( Aff ( R )) is uniformly locally stochastically -H¨older continuous, iffor any finite planar graphs G and G ′ in G (cid:0) Aff ( R ) (cid:1) such that G G ′ and for anyfamily of facial loops ( c F ) F ∈ F b of G , ( h ( l c F ,T )) F ∈ F b has the same law under µ G asunder µ G ′ , then there exists a unique stochastically continuous random holonomyfield µ on the plane such that for any finite planar graph G in G (cid:0) Aff ( R ) (cid:1) , for anyrooted spanning tree T and for any family of facial loops ( c F ) F ∈ F b of G , the law of ( h ( l c F ,T )) F ∈ F b is the same under µ as under µ G . Proof. For any finite planar graph G in G (cid:0) Aff ( R ) (cid:1) and any vertex v of G ,there exists a natural measurable function from Hom (cid:0) RL v ( G ) , G ∨ (cid:1) to the multi-plicative functions M ult P ( G ) ( L v ( G ) , G ): we can transport any measure from thefirst space to the second. Using the freeness of the generating families l c F ,T , themultiplicity property of random holonomy fields and Proposition 1.39, we can ex-tend µ G as a gauge-invariant random field on P ( G ). This gauge-invariant randomfield does not depend on the choice of v . An application of Proposition 1.40 andLemma 2.20 allows us to construct the desired µ . The uniqueness of µ is a conse-quence of Proposition 5.15. (cid:3) art 2 Construction of Planar MarkovianHolonomy Fields HAPTER 6 Braids and Probabilities I: an Algebraic Point ofView and Ginite Random Sequences For any finite planar graph G , we have constructed in the last section a set ofgenerating family of facial lassos of G : what is the transformation which sends onegenerating family to an other? It has to be noticed that, as soon as the root ofthe spanning tree is chosen, for any generating family of lassos constructed in thelast section, their product, up to some suitable permutation, is always equal to thesame loop. This remark and Artin’s theorem, Theorem 6.6, motivate the study ofthe group of braids. A geometric definition of the braid group was given in the introduction. Onecan also define the braid group using a generator-relation presentation. Let n bean integer greater than 2. Definition . The braid group with n strands B n is the group with thefollowing presentation: (cid:28)(cid:0) β i (cid:1) n − i =1 | ∀ i, j ∈ { , ..., n − } , | i − j | = 1 = ⇒ β i β j β i = β j β i β j | i − j | > ⇒ β i β j = β j β i (cid:29) . The elements ( β i ) n − i =1 we defined in the introduction satisfy the braid grouprelations. An example of the first relation between β i and β j when | i − j | = 1 isgiven in Figure 1.This presentation of the braid group is not intuitive, yet it allows us to recallsome natural actions of the braid group B n : one on the free group of rank n andone on G n . Let F n be the free group of rank n generated by e , ..., e n and let G beany group. = Figure 1. The braid relation 512 6. BRAIDS AND PROBABILITIES I Definition . The natural action of B n on F n is given by: β i e i = e i +1 ,β i e i +1 = e i +1 e i e − i +1 ,β i e j = e j , for any j / ∈ { i, i + 1 } . There exists a diagrammatic way to compute the action: one puts e , ..., e n at the bottom of a diagram representing β , then propagates these e , ..., e n in thediagram from the bottom to the top with the rule that, at each crossing, the valueon the string which is behind does not change and the value on the upper string isconjugated by the value of the other so that the product from right to left remainsunchanged. At the end one gets a n -uple ( f , ..., f n ) at the top of the diagram: thebraid sends e i on f i . Definition . The natural action of B n on G n is given by: β i • ( x , ..., x i − , x i , x i +1 , ..., x n ) = ( x , ..., x i − , x i x i +1 x − i , x i , ..., x n ) , (6.1)for any integer i ∈ { , ..., n − } and n -tuple ( x i ) ni =1 in G n .There exists also a diagrammatic way to compute the action: one puts x , ..., x n at the upper part of a diagram representing β , then propagates these x , ..., x n in the diagram from the top to the bottom with the rule that, at each crossing, thevalue on the string which is behind does not change and the value on the upperstring is conjugated by the value of the other so that the product from left to rightremains unchanged. At the end one gets a n -uple ( y , ..., y n ) at the top of thediagram: the braid sends ( x , ..., x n ) on ( y , ..., y n ).Let h be a G -valued multiplicative function on the free group. This meansthat h (cid:0) x − (cid:1) = h ( x ) − and h ( xy ) = h ( y ) h ( x ) for any x and y in F n . For any n -uple ( f , ..., f n ) of elements of F n , we define h ( f , ..., f n ) = ( h ( f ) , ..., h ( f n )). Thefollowing lemma shows how both actions are linked: it is a consequence of thediagrammatic formulation of both actions. Lemma . For any braid β ∈ B n , h ( β • ( e , ..., e n )) = β − • h ( e , ..., e n ) . With the n -diagrams picture in mind, it is obvious that the application, whichsends a braid on the permutation obtained by erasing the information at eachcrossing, is a homomorphism: it is the one which sends β i on the transposition( i, i + 1) for any integer i ∈ { , ..., n − } . Lemma . The operation of erasing the information at each crossing inducesa natural homomorphism from B n to S n . We will denote the image of β by σ β . For any braid β with n strands, the action a β of β on F n is an automorphismof F n : there exists a morphism from B n in A ut ( F n ) which is moreover injective.In [ Art47 ] and [ Art25 ], Artin gave a sufficient and necessary condition for anautomorphism of F n to be the induced action of a braid in B n . Theorem . An automorphism a of F n is the induced action of a braid in B n if and only if the two following conditions hold: Conjugacy property: for any i in { , ..., n } , a ( e i ) is in the same conjugacyclass as one of the elements of ( e j ) nj =1 . .3. BRAIDS AND FINITE SEQUENCE OF RANDOM VARIABLES 53 Product invariance: a ( e n ... e ) = e n ... e . Remark . Let β be a braid in B n and a β the induced action on F n . Foreach i in { , ..., n } , a β ( e i ) is conjugated to e σ β ( i ) and this property characterizes σ β .Let G = ( V , E , F ) be a finite planar graph, v be a vertex of G , T and T ′ be twospanning trees of G rooted at v , ( c F ) F ∈ F b and ( c ′ F ) F ∈ F b be two families of orientedfacial loops oriented anti-clockwise. There exist two freely generating families ofthe group of reduced loops of G associated with ( c F ) F ∈ F b and ( c ′ F ) F ∈ F b . UsingArtin theorem, we can characterize the transformation which sends one family onthe other. Proposition . There exists an enumeration of the bounded faces ( F i ) F b i =1 and a braid β in B F b such that: β • (cid:0) l c Fi ,T (cid:1) F b i =1 = (cid:0) l c ′ Fσ ( i ) ,T ′ (cid:1) F b i =1 , where σ = σ β and where β is seen as acting on the free group generated by (cid:0) l c Fi ,T (cid:1) F b i =1 . Proof. For any bounded face F of G , the first part of Proposition 5.12 assertsthat l c F ,T and l c ′ F ,T ′ are facial lassos based at v whose meanders represent thefacial cycle ∂F oriented anti-clockwise. By Lemma 5.2, we deduce that l c ′ F ,T ′ isconjugated to l c F ,T in RL v ( G ). Besides, thanks to Lemma 5.13, we can find anenumeration of the bounded faces ( F i ) F b i =1 and a permutation σ of { , ..., F b } suchthat: (1) l c Fn ,T l c Fn − ,T ... l c F ,T = l ∞ , (2) l c ′ Fσ ( n ) ,T ′ l c ′ Fσ ( n − ,T ′ ... l c ′ Fσ (1) ,T ′ = l ∞ , in RL v ( G ), where l ∞ is the facial loop based at v , turning anti-clockwise, represent-ing the non-oriented facial cycle ∂F ∞ . Besides, Proposition 5.12 tells us that both( l c F ) F ∈ F b and ( l c ′ F ) F ∈ F b are free families of generators of the free group RL v ( G ).A natural automorphism of RL v ( G ) is defined by: ∀ i ∈ { , ..., F b } , a ( l c Fi ,T ) = l c ′ Fσ ( i ) ,T ′ . This automorphism of free group satisfies the conditions of Artin’s theorem given inTheorem 6.6. There exists a braid β such that a is equal to a β , the action inducedby β on the free group RL v ( G ) with free generators ( l c F ,T ) F ∈ F b . Using Remark6.7, it is straightforward to see that σ is equal to σ β . (cid:3) In the last section, the transformations between families of loops of the form( l c F ,T ) F ∈ F b have been characterized. In the context of random holonomy fields, arandom variable is associated with any loop: it is natural to study the action of thebraid groups on finite sequence of random variables. When one has to deal withnon-commutative random variables (i.e. random variables in a non-commutativegroup), this action is in some sense more appropriate than the symmetrical groupaction which is often studied in the mathematical literature. This leads to a theoryof braidability which is more efficient than the exchangeability concept for sequencesof random variables in a non-commutative group. Let n be an integer strictly greaterthan 1 and G be an arbitrary topological group. Definition . The braid group B n acts on the set of n -tuple of G -valuedrandom variables according to the formula: β i • ( X , ..., X i − , X i , X i +1 , ..., X n ) = ( X , ..., X i − , X i X i +1 X − i , X i , ..., X n )(6.2)for any i ∈ { , ..., n − } .Recall the notation σ β which was defined in Lemma 6.5. Definition . Let ( X , ..., X n ) be a finite sequence of G -valued randomvariables. It is purely invariant by braids if for any braid β ∈ B n one has theequality in law: β • ( X , ..., X n ) = σ β • ( X , ..., X n ) , where σ • ( X , ..., X n ) = (cid:0) X σ − (1) , ..., X σ − ( n ) (cid:1) for any permutation σ ∈ S n .It is invariant by braids if for any braid β ∈ B n , one has the equality in law: β • ( X , ..., X n ) = ( X , ..., X n ) . Let us recall that if m is a probability measure on G , the support of m , denotedby Supp ( m ), is the smallest closed subset of G of measure 1 for m . The closure ofthe subgroup generated by the support of m is denoted by H m . If X is a G -valuedrandom variable and m its law, we define Supp ( X ) = Supp ( m ) and H X = H m . Let T be a finite index set such that T ≥ Definition . Let ( X t ) t ∈ T be a sequence of G -valued random variables.We say that ( X t ) t ∈ T is auto-invariant by conjugation if for any different elements i and j in T and for any g ∈ Supp ( X j ), we have the equality in law: gX i g − = X i . (6.3)This definition can be extended to collections of measures on G .The first result on random sequences which are purely invariant by braids isthe following proposition. Proposition . A finite sequence of independent G -valued random variablesis auto-invariant by conjugation if and only if it is purely invariant by braids. Proof. Let ( X , ..., X n ) be a finite sequence of G -valued random variableswhich are independent. Let us suppose that ( X , ..., X n ) is auto-invariant by con-jugation. Since β σ β is a morphism and using the independence of the variables,we just have to show that ( X , X − X X ) and ( X , X ) have the same law. Thisresult follows from the independence of the variables X and X and from theinvariance by conjugation of the law of X by any element g in the support of X .Now, let us suppose instead that ( X , ..., X n ) is purely invariant by braids. Let i < j be two integers in { , ..., n } . Let β ( i,j ) be the braid defined by: β ( i,j ) = β − i ...β − j − β j − ...β i . An example of such a braid is shown in Figure 2. By considering only the i th and j th positions in the equality in law β ( i,j ) • ( X , ..., X n ) = σ β i,j • ( X , ..., X n ), we getthe following equality in law:( X i , X j ) = ( X i , X i X j X − i ) . By disintegration and using the independence of the variables, one gets the desiredresult. (cid:3) .3. BRAIDS AND FINITE SEQUENCE OF RANDOM VARIABLES 55 i Figure 2. The braid β ( i,j ) .The proof of Proposition 6.12 is straightforward, but looking at the followingequality in law: ( X − X X , X − X − X X X ) = ( X , X ) , where ( X , X ) is anauto-invariant by conjugation couple of random variables, one can see that it givesidentities which, at first glance, do not seem trivial. A last remark to be made aboutProposition 6.12 is that there exist finite sequences of non-independent G -valuedrandom variables which are purely invariant by braids.HAPTER 7 Planar Yang-Mills Fields In this chapter, we construct a family of planar Markovian holonomy fields:the planar Yang-Mills fields. First, we construct the pure planar Yang-Mills fields.Then, in Section 7.2, we generalize the construction in order to construct all planarYang-Mills fields. In order to construct pure planar Yang Mills fields, given any L´evy process Y which is invariant by conjugation by G , we define, in Proposition 7.2, for anyfinite planar graph G , an random holonomy field on P ( G ) associated to Y . InPropositions 7.3 and 7.5, we show that these random holonomy fields allow us todefine a family of random holonomy fields on R which is a strong planar Markovianholonomy field. In Section 7.2, we weaken the condition on the Lvy process by usingour results about the extension of the structure group. First of all, let us recall thedefinition of L´evy processes. Definition . A L´evy process ( Z t ) t ≥ is a random c`adl`ag process from R + to G , with independent and stationary right increments. This means:(1) ∀ ≤ t < ... < t n , (cid:0) Z − t i − Z t i (cid:1) ni =1 are independent,(2) ∀ ≤ s < t , Z − s Z t has the same law as Z t − s .We say that ( Z t ) t ≥ is invariant by conjugation by G , or conjugation-invariant, ifand only if for any g ∈ G , the process ( g − Z t g ) t ≥ has the same law as ( Z t ) t ≥ .There is a correspondence between continuous semi-groups of convolution ofprobability measures starting from the Dirac measure on the neutral element of G and L´evy processes. Let us consider Y = ( Y t ) t ≥ a conjugation-invariant L´evyprocess on G which is fixed until the end of Section 7.1. Proposition . Let G be a finite planar graph, let vol be a measure of area.There exists a unique random holonomy field E Y, G vol on P ( G ) , whose weight is equalto , such that for any rooted spanning tree T of G , any family ( c F ) F ∈ F b of facialloops of G , each oriented anti-clockwise, under E Y, G vol : (1) the random variables (cid:0) h ( l c F ,T ) (cid:1) F ∈ F b are independent, (2) for any F ∈ F b , h ( l c F ,T ) has the same law as Y vol ( F ) .The family (cid:16) E Y, G vol (cid:17) G ,vol is the discrete planar Yang-Mills field associated with Y . Proof. Let G be a finite planar graph, let vol be a measure of area. Forany positive real t , let us denote by m t the law of Y t . For any rooted spanningtree T and any family ( c F ) F ∈ F b of facial loops oriented anti-clockwise, we define 578 7. PLANAR YANG-MILLS FIELDS the measure E Y, G vol,T, ( c F ) F ∈ F b on (cid:0) M ult ( P ( G ) , G ) , B (cid:1) as the unique gauge-invariantprobability measure such that, under E Y, G vol,T, ( c F ) F ∈ F b :(1) the random variables (cid:0) h ( l c F ,T ) (cid:1) F ∈ F b are independent,(2) for any F ∈ F b , h ( l c F ,T ) has the same law as Y vol ( F ) ,where we remind the reader that the loops l c F ,T where defined in Definition 5.11.Since ⊗ F ∈ F b m vol ( F ) is invariant by diagonal conjugation, by applying the first partof Proposition 5.16 we see that the definition makes sense. We will show that theprobability measure E Y, G vol,T, ( c F ) F ∈ F b neither depends on the choice of T , nor on thechoice of ( c F ) F ∈ F b . Thanks to the uniqueness property in this last definition, wehave to prove that given another rooted spanning tree T ′ and another family offacial loops ( c ′ F ) F ∈ F b oriented anti-clockwise, under E Y, G vol,T, ( c F ) F ∈ F b , (cid:0) h ( l c ′ F ,T ′ ) (cid:1) F ∈ F b has the same law as (cid:0) h ( l c F ,T ) (cid:1) F ∈ F b .First of all, let us prove that one can suppose that T and T ′ are rootedat the same vertex v of G . Let v be the root of T , let v ′ be a vertex of G and let us define the rooted spanning tree ˜ T as the tree T rooted at v ′ . Whenwe change the root of T from v to v ′ we conjugate every of the l c F ,T by thesame path [ v ′ , v ] T . By Remark 1.38, since E Y, G vol,T, ( c F ) F ∈ F b is gauge-invariant, un-der E Y, G vol,T, ( c F ) F ∈ F b , (cid:0) h ( l c F , ˜ T ) (cid:1) F ∈ F b has the same law as (cid:0) h ( l c F ,T ) (cid:1) F ∈ F b , namely ⊗ F ∈ F b m vol ( F ) : E Y, G vol,T, ( c F ) F ∈ F b = E Y, G vol, ˜ T , ( c F ) F ∈ F b .Now let us assume that T and T ′ are rooted at the same vertex. By Proposition6.8, there exists an enumeration ( F i ) F b i =1 of the bounded faces of G , a braid β in B F b such that: β • (cid:0) l c Fi ,T (cid:1) F b i =1 = (cid:18) l c ′ Fσβ ( i ) ,T ′ (cid:19) F b i =1 . Using Lemma 6.4, h (cid:18) β • (cid:0) l c Fi ,T (cid:1) F b i =1 (cid:19) = β − • (cid:0) h ( l c Fi ,T ) (cid:1) F b i =1 , and thus: β − • (cid:0) h ( l c Fi ,T ) (cid:1) F b i =1 = σ β − • (cid:16) h ( l c ′ Fi ,T ′ ) (cid:17) F b i =1 . Applying the Proposition 6.12, under E Y, G vol,T, ( c F ) F ∈ F b , the following equality in lawholds: β − • (cid:0) h ( l c Fi ,T ) (cid:1) F b i =1 = σ β − • (cid:0) h ( l c Fi ,T ) (cid:1) F b i =1 . From this, we get the equality in law under E Y, G vol,T, ( c F ) F ∈ F b : (cid:16) h ( l c ′ Fi ,T ′ ) (cid:17) F b i =1 = (cid:16) h ( l c Fi ,T ) (cid:17) F b i =1 . (cid:3) This proposition allows us not to have to choose a special rooted tree for eachgraph in order to construct planar Yang-Mills fields. More importantly, it will allowus to show the independence property and the area-preserving homeomorphisminvariance of the family of random holonomy fields which we will construct thanksto Proposition 5.16. .1. CONSTRUCTION OF PURE PLANAR YANG-MILLS FIELDS 59 Proposition . There exists a unique family of gauge-invariant stochasti-cally continuous random holonomy fields (cid:0) E Yvol (cid:1) vol , whose weight is equal to , suchthat for any measure of area vol , for any finite planar graph G , for any rooted span-ning tree T of G and any family of facial loops ( c F ) F ∈ F b oriented anti-clockwise,under E Yvol : (1) the random variables ( h ( l c F ,T )) F ∈ F b are independent, (2) for any F ∈ F b , h ( l c F ,T ) has the same law as Y vol ( F ) .The family (cid:0) E Yvol (cid:1) vol is the planar Yang-Mills field associated with ( Y t ) t ≥ . In order to prove this result, we will need the following statement, from [ L´ev10 ],which allows us to bound the distance of a L´evy process to the neutral element. Proposition . There exists K > such that E (cid:2) d G (1 , Y t ) (cid:3) ≤ K √ t for any t ≥ . Proof of Proposition 7.3. Let vol be a measure of area on the plane. Wewill apply Proposition 5.16 to the family of measures (cid:0) E Y, G vol (cid:1) G ∈G ( Aff ( R )) . Then wewill study the restriction to general finite planar graphs of the random holonomyfield that we will have defined. In order to do all this, we have to prove a compat-ibility condition and a uniform locally stochastically -H¨older continuity propertyfor the family (cid:0) E Y, G vol (cid:1) G ∈G ( Aff ( R )) . Compatibility condition: Let G and G be two graphs in G (cid:0) Aff ( R ) (cid:1) suchthat G G . Let us consider m a vertex of G and G . Using Proposition 7.2, itis enough to show that G satisfies the following property:( H ) there exists a family of facial loops ( c F ) F ∈ F b oriented anti-clockwise anda spanning tree T of G rooted at m , such that under E Y, G vol :1. the random variables (cid:0) h ( l c F ,T ) (cid:1) F ∈ F b are independent,2. for any F ∈ F b , h ( l c F ,T ) has the same law as Y vol ( F ) .We show this by an induction argument on the finite set (cid:2) G , G (cid:3) which isequal to (cid:8) G , G G G (cid:9) , endowed with the partial order . It is clearlytrue that ( H ) holds for G = G . Consider a finite planar graph G in (cid:2) G , G (cid:3) satisfying ( H ), we will show that there exists G ′ ∈ (cid:2) G , G (cid:2) for which ( H ) is stillvalid. Thanks to Proposition 7.2, property ( H ) holds for G for any family of facialloops ( c F ) F ∈ F b oriented anti-clockwise and any choice of spanning tree T rooted at m . Since G G , at least one of the following assertions is true:(1) there exist an edge of G , e , and a vertex v of G of degree two such that v ∈ e (cid:0) (0 , (cid:1) ,(2) there exists a face F of G , bounded or not, such that the restriction of G to F has a unique face F and ∂F , oriented anti-clockwise, contains asequence of the form ee − with the interior of e included in F ,(3) there exists a face F of G which contains more than one face of G .Let us consider the three possibilities.(1) Let us consider a family of facial loops ( c F ) F ∈ F b for G , oriented anti-clockwise, none of which is based at v , and a choice of spanning tree T of G rootedat m . Let e and e be the two edges of G such that e = e e and e = v . Weconsider G ′ , the graph defined by:( V ′ , E ′ , F ′ ) = (cid:0) V \ { v } , E \ (cid:8) e ± , e ± (cid:9) ∪ (cid:8) ( e e ) ± (cid:9) , F (cid:1) . By construction G ′ ∈ (cid:2) G , G (cid:2) . Besides, ( c F ) F ∈ F ′ b is still a family of facial loops for G ′ oriented anti-clockwise and T ′ = (cid:0) T \ (cid:8) e ± , e ± (cid:9) (cid:1) ∪ (cid:8) ( e e ) ± (cid:9) is a spanningtree of G ′ rooted at m . It is now obvious that G ′ satisfies property ( H ) with thechoices of ( c F ) F ∈ F ′ b and T ′ .(2) We will consider that F is bounded, the unbounded case is similar. Inthis case, let v be the vertex of e of degree 1 and define F ′ = F ∪ e (cid:0) (0 , (cid:1) ∪ { v } .Consider any family of facial loops for G oriented anti-clockwise, ( c F ) F ∈ F b , suchthat c F = v . Let us choose any spanning tree of G rooted at m , T . We consider G ′ , the graph defined by: (cid:0) V ′ , E ′ , F ′ (cid:1) = (cid:0) V \ v, E \ { e, e − } , ( F \ F ) ∪ F ′ (cid:1) . The spanning tree T of G must include the unoriented edge (cid:8) e, e − (cid:9) in order tocover v , thus we can define T ′ = T \ { e, e − } . The facial loop c F contains thesequence ee − . We define c ′ F ′ from c F by removing this sequence. For any otherface F ∈ F ′ , we set c ′ F = c F . For any face F ∈ F ′ b , using the identification between F and F ′ , l c F ,T = l c ′ F ,T ′ in RL m ( G ), and by Remark 1.16, h ( l c F ,T ) = h ( l c ′ F ,T ′ ).The graph G ′ satisfies property ( H ) with the choices of ( c ′ F ′ ) F ′ ∈ F ′ b and T ′ .(3) We will study this case under the hypothesis that F is bounded, the un-bounded case being easier. The key point will be the semigroup property satisfiedby the marginal distributions of the L´evy process Y . Let F r and F l be two facesof G contained in F and adjacent, sharing an edge e on their boundaries. We canfind a facial loop oriented anti-clockwise representing the boundary of F r (resp. F l )of the form c F r = e ...e n e (resp. c F l = e − e ′ ...e ′ m ). Let F r,l = F r ∪ F l ∪ e (cid:0) (0 , (cid:1) .We complete the family ( c F r , c F l ) in order to have a family of facial loops ( c F ) F ∈ F b oriented anti-clockwise for G . Let us consider G ′ , the graph defined by:( V ′ , E ′ , F ′ ) = (cid:16) V , E \ (cid:8) e, e − (cid:9) , ( F \ { F r , F l } ) ∪ F r,l (cid:17) . It is still a finite planar graph. Let us consider T any spanning tree of G ′ rooted at m : it is also a spanning tree of G rooted at m . Let c ′ F r,l = e ...e n e ′ ...e ′ m . For anyother face F ′ of G ′ different from F r,l , F ′ is a face of G and we set c ′ F ′ = c F ′ . Oncethese choices made, it needs only a simple verification to check that the followingequalities hold in RL m ( G ): l c ′ Fr,l ,T = l c Fr ,T l c Fl ,T ,l c ′ F ′ ,T = l c F ′ ,T , ∀ F ′ ∈ F ′ b , F ′ = F r,l . Using the multiplicativity of h : h ( l c ′ Fr,l ,T ) = h ( l c Fl ,T ) h ( l c Fr ,T ) ,h ( l c ′ F ′ ,T ) = h ( l c F ′ ,T ) , ∀ F ′ ∈ F ′ b , F ′ = F r,l . Let us recall that under E Y, G vol ,(1) the random variables ( h ( l c F ,T )) F ∈ F b are independent,(2) for any F ∈ F b , h ( l c F ,T ) has the same law as Y vol ( F ) .Using the semigroup property of the marginal distributions of the process Y , wecan conclude that G ′ satisfies ( H ) with the choices of ( c ′ F ′ ) F ′ ∈ F ′ b and T .By descending induction, it follows that G satisfies property ( H ). Let us provethe uniform -H¨older continuity. .1. CONSTRUCTION OF PURE PLANAR YANG-MILLS FIELDS 61 Uniform -H¨older continuity: Let G be a finite planar graph with piecewiseaffine edges. Let l be a simple loop in G bounding a disk D . A consequence of whatwe have just seen is that the law of h ( l ) under E Y, G vol is the same as under E Y, G ( l ) vol ,where G ( l ) is the graph containing only the edge l (see Example 2.3). Thus: Z M ult ( P ( G ) ,G ) d G (cid:0) , h ( l ) (cid:1) E Y, G vol ( dh ) = Z M ult ( P ( G ( l )) ,G ) d G (cid:0) , h ( l ) (cid:1) E Y, G ( l ) vol ( dh )= E (cid:2) d G (cid:0) , Y vol ( D ) (cid:1)(cid:3) ≤ K p vol ( D ) , where the last inequality comes from Proposition 7.4 and where K depends onlyon G . The family ( E Y, G vol ) G ∈G ( Aff ( R )) is uniformly locally stochastically -H¨oldercontinuous.Thus, Proposition 5.16 can be applied in order to construct a stochasticallycontinuous random holonomy field E Yvol such that for any finite planar graph G ∈G ( Aff (cid:0) R (cid:1) ), for any rooted spanning tree T of G and any family of facial loops( c F ) F ∈ F b oriented anti-clockwise, under E Yvol :(1) the random variables (cid:0) h ( l c F ,T ) (cid:1) F ∈ F b are independent,(2) for any F ∈ F b , h ( l c F ,T ) has the same law as Y vol ( F ) .It remains to prove that this property is true for any finite planar graph, notnecessarily with piecewise affine edges. Let ( m t ) t ∈ R + be the continuous semi-group of convolution associated with ( Y t ) t ≥ . Let G = ( V , E , F ) be a finite pla-nar graph, let T be a rooted spanning tree and let (cid:0) c F (cid:1) F ∈ F b be a family of facialloops oriented anti-clockwise. Let us consider a sequence of finite planar graphs (cid:0) G n = ( V n , E n , F n ) (cid:1) n ∈ N in G (cid:0) Aff ( R ) (cid:1) and ( ψ n ) n ∈ N a sequence of orientation-preserving homeomorphisms which satisfy the conditions of Theorem 2.21. Forany integer n , ( ψ n ( c F )) F ∈ F b is a family of facial loops for G n which is orientedanti-clockwise and ψ n ( T ) is a spanning tree of G n . Using the discussion we hadbefore, the law of (cid:0) h ( l ψ n ( c F ) ,ψ n ( T ) ) (cid:1) F ∈ F b under E Yvol is N F ∈ F b m vol ( ψ n ( F )) . As for anyedge e ∈ E , ( ψ n ( e )) n ≥ converges to e for the convergence with fixed endpoints, forany face F ∈ F b , one has l ψ n ( c F ) ,ψ n ( T ) −→ n →∞ l c F ,T for the fixed endpoints conver-gence. Besides, using condition 4 of Theorem 2.21 and the continuity of ( m t ) t ∈ R + , N F ∈ F b m vol ( ψ n ( F )) −→ n →∞ N F ∈ F b m vol ( F ) . Since E Yvol is stochastically continuous, under E Yvol , the law of (cid:0) h ( l c F ,T ) (cid:1) F ∈ F b is N F ∈ F b m vol ( F ) . (cid:3) Let us remark that, in the latest argument, we actually proved that the family (cid:0) E Y, G vol (cid:1) G ,vol is continuously area-dependent. For any L´evy process which is invari-ant by conjugation, we have constructed a family of gauge-invariant stochasticallycontinuous random holonomy fields. In the following, we will show that this familyis a strong planar Markovian holonomy field. Proposition . The family of random holonomy fields (cid:0) E Yvol (cid:1) vol is a con-structible stochastically continuous strong planar Markovian holonomy field. Proof. We have already shown that the family (cid:0) E Y, G vol (cid:1) G ,vol satisfies the Axiom DP , is continuously area-dependent and locally stochastically -H¨older continu-ous. By Theorem 3.11, it remains to check that (cid:0) E Y, G vol (cid:1) G ,vol satisfies the threeAxioms DP , DP and DP in Definition 3.4. Besides, (cid:0) E Y, G vol (cid:1) G ,vol is stochastically continuous in law: using a continuity ar-gument, it is enough to show that wDP holds instead of DP . Let us give brieflythe arguments which allow us to do so: first of all, using Theorem 2.21, we seethat it is enough to consider graphs with piecewise affine edges. Let us supposethat the Axiom wDP holds, let us consider G a finite planar graph with piece-wise affine edges and two loops l and l in P ( G ) such that Int ( l ) ∩ Int ( l ) = ∅ .The only interesting case is when Int ( l ) ∩ Int ( l ) = ∅ : let us consider a point v in this intersection. Using Remark 1.38, it is enough to prove that for any familyof loops ( l i ) ni =1 (resp. ( l i ) mi =1 ) in Int ( l ) (resp. in Int ( l )) based at v , (cid:0) h ( l i ) (cid:1) ni =1 is I -independent of (cid:0) h ( l i ) (cid:1) mi =1 under the measure E Y, G vol . Let us consider such familiesof loops ( l i ) ni =1 and ( l i ) mi =1 . One can always approximate G by a finite planar graph G ′ with piecewise affine edges such that there exist l ′ and l ′ two loops in G ′ and p a path in G ′ such that:(1) Int ( l ′ ) ∩ Int ( l ′ ) = ∅ ,(2) for any loop l in Int ( l ) based at v , there exists a loop in Int ( l ′ ) whichapproximates l for the convergence with fixed endpoints,(3) for any loop l in Int ( l ) based at v , there exist a loop in Int ( l ′ ), denoted by l ′ such that pl ′ p − approximates l for the convergence with fixed endpoints.Let us consider the graph G ′ , the two loops l ′ and l ′ and the path p given bythe last assertion. We can approximate ( l i ) ni =1 and ( l i ) mi =1 by two families ( l ′ i ) ni =1 and ( pl ′ i p − ) mi =1 such that the first one is in Int ( l ′ ) and ( l ′ i ) mi =1 is in Int ( l ′ ). The I -independence of ( l i ) ni =1 and ( l i ) mi =1 under the measure E Y, G vol would be a consequenceof the I -independence of ( l ′ i ) ni =1 and ( pl ′ i p − ) mi =1 under E Y, G ′ vol , which is equivalentto the the I -independence of ( l ′ i ) ni =1 and ( l ′ i ) mi =1 under E Y, G ′ vol . Using Remark 1.32,this is equivalent to the independence of ( l ′ i ) ni =1 and ( l ′ i ) mi =1 under E Y, G ′ vol which isgranted since we supposed that the Axiom wDP holds.Let us prove that (cid:0) E Y, G vol (cid:1) G ,vol satisfies the three Axioms DP , wDP and DP . DP : Consider vol and vol ′ two measures of area on R , G and G ′ two finiteplanar graphs and ψ a homeomorphism which preserves the orientation. Let ussuppose that ψ ( G ) = G ′ and for any F ∈ F b , vol ( F ) = vol ′ ( ψ ( F )). Let ( c ′ F ) F ∈ F ′ b be a family of facial loops oriented anti-clockwise for G ′ and let T ′ be a rootedspanning tree of G ′ . We consider (cid:0) c F = ψ − (cid:0) c ′ ψ ( F ) (cid:1)(cid:1) F ∈ F b and T = ψ − ( T ′ ). Thefamily ( c F ) F ∈ F b is a family of facial loops for G which are oriented anti-clockwiseand T is a rooted spanning tree of G . Recall the notations used in the proof ofProposition 7.2. By construction, we have the equality: E Y, G ′ vol ′ ,T ′ , ( c ′ F ) F ∈ F ′ b ◦ ψ − = E Y, G vol,T, ( c F ) F ∈ F b , where we denoted also by ψ the induced application from M ult ( P ( G ′ ) , G ) to M ult ( P ( G ) , G ) induced by the homeomorphism ψ . Using the Proposition 7.2,we get E Y, G ′ vol ′ ◦ ψ − = E Y, G vol . wDP : Let vol be a measure of area on R , G = ( V , E , F ) be a finite planargraph in G (cid:0) Aff (cid:0) R (cid:1)(cid:1) , m be a vertex of G and L and L be two simple loops in P ( G ) whose closure of the interiors are disjoint. As an application of Proposition5.14, we can consider T a spanning tree rooted at m , such that for any family offacial loops ( c F ) F ∈ F b oriented anti-clockwise: .1. CONSTRUCTION OF PURE PLANAR YANG-MILLS FIELDS 63 (1) for every loop l in P ( G ) inside Int ( L ), h [ m, l ] T l [ m, l ] − T i ≃ is a product in RL m ( G ) of elements of n l ± c F ,T ; F ∈ F b , F ⊂ Int ( L ) o ,(2) for every loop l in P ( G ) inside Int ( L ), h [ m, l ] T l [ m, l ] − T i ≃ is a product in RL m ( G ) of elements of n l ± c F ,T ; F ∈ F b , F ⊂ Int ( L ) o .Let us consider ( p i ) ni =1 (resp. ( p ′ j ) n ′ j =1 ), some paths in P ( G ) which are inside Int ( L ) (resp. Int ( L )). Recall the definition given by Equality (1.5). For anycontinuous function f (resp. f ) defined on G n (resp. on G n ′ ), using the gauge-invariance, E Y, G vol h f (( h ( p i )) ni =1 ) f (cid:16)(cid:0) h ( p ′ j ) (cid:1) n ′ j =1 (cid:17)i is equal to: E Y, G vol (cid:20) ˆ f J p ,...,pn ( h ( p ) , ..., h ( p n )) ˆ f J p ′ ,...,p ′ n ′ ( h ( p ′ ) , ..., h ( p ′ n ′ )) (cid:21) . Thus, it is also equal to: E Y, G vol (cid:20) ˆ f J p ,...,pn (cid:16)(cid:16) h (˜ l i ) (cid:17) ni =1 (cid:17) ˆ f J p ′ ,...,p ′ n ′ (cid:18)(cid:16) h (˜ l ′ i ) (cid:17) n ′ i =1 (cid:19)(cid:21) . where, for any i ∈ { , ..., n } , ˜ l i = [ m, p i ] T p i [ m, p i ] − T and for any i ∈ { , ..., n ′ } ,˜ l ′ i = [ m, p ′ i ] T p ′ i [ m, p ′ i ] − T . Recall the form of T given in the proof of Proposition5.14: this implies that there exist ( l i ) ni =1 some loops in Int ( L ) and ( l ′ i ) n ′ i =1 someloops in Int ( L ) such that for any i ∈ { , ..., n } and any j ∈ { , ..., n ′ } ,˜ l i = [ m, l i ] T l i [ m, l i ] − T , ˜ l ′ j = [ m, l ′ j ] T l ′ j [ m, l ′ j ] − T , in RL m ( G ). Using the properties satisfied by T : σ (cid:16)(cid:16) h (˜ l i ) (cid:17) ni =1 (cid:17) ⊂ σ (cid:16)n h ( l c F ,T ); F ∈ F b , F ⊂ Int ( L ) o(cid:17) ,σ (cid:18)(cid:16) h (˜ l ′ i ) (cid:17) n ′ i =1 (cid:19) ⊂ σ (cid:16)n h ( l c F ,T ); F ∈ F b , F ⊂ Int ( L ) o(cid:17) . We recall that Int ( L ) ∩ Int ( L ) = ∅ , thus the two σ -fields: σ (cid:16)n h ( l c F ,T ); F ∈ F b , F ⊂ Int ( L ) o(cid:17) ,σ (cid:16)n h ( l c F ,T ); F ∈ F b , F ⊂ Int ( L ) o(cid:17) are independent under E Y, G vol . Thus E Y, G vol h f (( h ( p i )) ni =1 ) f (cid:16)(cid:0) h ( p ′ j ) (cid:1) n ′ j =1 (cid:17)i is equalto: E Y, G vol h ˆ f J p ,...,pn (cid:16)(cid:16) h (˜ l i ) (cid:17) ni =1 (cid:17)i E Y, G vol (cid:20) ˆ f J p ′ ,...,p ′ n ′ (cid:18)(cid:16) h (˜ l ′ i ) (cid:17) n ′ i =1 (cid:19)(cid:21) , which is equal to E Y, G vol [ f (( h ( p i )) ni =1 )] E Y, G vol h f (cid:16) ( h ( p ′ j )) n ′ j =1 (cid:17)i : the axiom wDP issatisfied. DP : Let l be a simple loop, let vol and vol ′ be two measures of area on R which are equal in the interior of l . Let G be a finite planar graph included in Int ( l ). The bounded faces of G are in the interior of l thus for any bounded face F of G , vol ( F ) = vol ′ ( F ). By definition, it is clear that E Y, G vol = E Y, G vol ′ . We have proved all the conditions on (cid:0) E Y, G vol (cid:1) G ,vol we needed in order to applyTheorem 3.11. The family of holonomy fields (cid:0) E Yvol (cid:1) vol is a constructible stochasti-cally continuous strong planar Markovian holonomy field. (cid:3) For this new construction, we used the loop paradigm which links the mul-tiplicative functions on a set P and the pre-multiplicative functions on its set ofloops. The edge paradigm given by the Equation (1.1) can be used to give anexplicit formula for discrete planar Yang-Mills fields associated with a conjugationinvariant L´evy process with density. Proposition . Let us suppose that for any positive real t , Y t has a density Q t with respect to the Haar measure. Let (cid:0) E Y, G vol (cid:1) G ,vol be the discrete planar Yang-Mills field associated with Y . For any finite planar graph G and for any measureof area vol : E Y, G vol ( dh ) = Y F ∈ F b Q vol ( F ) (cid:0) h ( ∂F ) (cid:1) O e ∈ E + dh ( e ) , (7.1) where ∂F is the anti-clockwise oriented facial cycle associated with F , the notation Q vol ( F ) (cid:0) h ( ∂F ) (cid:1) means that we consider Q vol ( F ) (cid:0) h ( c ) (cid:1) where c represents ∂F (thisdoes not depend on the choice of c since Q vol ( F ) is invariant by conjugation) and N e ∈ E + dh ( e ) is the push-forward of N e ∈ E + dg e on M ult ( P, G ) by the edge paradigmidentification. It is independent of the choice of orientation E + . Recall the definition of L i,j in Definition 5.9. In order to make the proof simple,we will use the upcoming Theorem 9.6 which roughly asserts that a stochasticallycontinuous planar Markovian holonomy field is characterized by the law of therandom sequence ( h ( L n, )) n ∈ N . Proof. A slight modification of Section 4.3 in [ L´ev10 ] shows that: (cid:18) Y F ∈ F b Q vol ( F ) ( h ( ∂F )) O e ∈ E + dh ( e ) (cid:19) G ,vol (7.2)is a stochastically continuous in law discrete planar Markovian holonomy field.Let G = ( V , E , F ) be the planar graph N ∩ (cid:0) R + × [0 , (cid:1) . Let us consider E + an orientation of E . As an application of Theorem 9.6, we only have to checkthat for any positive real α , (cid:0) h ( L n, ) (cid:1) n ∈ N has the same law under E Y, G αdx as under Q F ∈ F b Q α (cid:0) h ( ∂F ) (cid:1) N e ∈ E + dh ( e ). Since the value of α will not matter we suppose that α equals to 1.By Proposition 7.2, under E Y, G dx , (cid:0) h ( L n, ) (cid:1) n ∈ N are i.i.d. random variables whichhave the same law as Y . It remains to prove that this property is true under Q F ∈ F b Q (cid:0) h ( ∂F ) (cid:1) N e ∈ E + dh ( e ).Under the probability law N e ∈ E + dh ( e ), (cid:0) h ( e ) (cid:1) e ∈ E + are i.i.d. and Haar dis-tributed. Using the multiplicativity property of random holonomy fields, for anyinteger n , h ( L n, ) is a product of elements of ( h ( e )) e ∈ E = ( h ( e )) e ∈ E + ∪ ( h ( e ) − ) e ∈ E + .An important remark is that for any integer n , the edge e rn, , defined in the Nota-tion 5.8, appears only once in the reduced decomposition of L n, and in no otherreduced decomposition of L m, with m = n . Applying Lemma 7.7, one has that un-der N e ∈ E + dh ( e ), (cid:0) h ( L n, ) (cid:1) n ∈ N are independent and each of them is a Haar randomvariable. Recall the notation ∂c i,j defined in Definition 5.9. Since Y is invariant .2. CONSTRUCTION OF GENERAL PLANAR YANG-MILLS FIELDS 65 by conjugation, for any bounded face F in F , there exists an integer n ∈ N suchthat Q ( ∂F ) = Q (cid:0) h ( ∂c n, ) (cid:1) = Q (cid:0) h ( L n, ) (cid:1) . Let f : G N → R be a measurablefunction, the following sequence of equality holds: Z M ult ( P ( G ) ,G ) f (cid:0) ( h ( L n, )) n ∈ N (cid:1) Y F ∈ F b Q ( h ( ∂F )) O e ∈ E + dh ( e )= Z M ult ( P ( G ) ,G ) f (cid:0) ( h ( L n, )) n ∈ N (cid:1) Y n ∈ N Q ( h ( ∂c n, )) O e ∈ E + dh ( e )= Z M ult ( P ( G ) ,G ) f (cid:0) ( h ( L n, )) n ∈ N (cid:1) Y n ∈ N Q ( h ( L n, )) O e ∈ E + dh ( e )= Z G N f (cid:0) ( g n ) n ∈ N (cid:1) O n ∈ N ( Q ( g n ) dg n ) , which is the assertion we had to prove. (cid:3) Lemma . Let ( α i ) ∞ i =1 be a sequence of independent G -valued random vari-ables which are Haar distributed. Let ( β j ) ∞ j =1 be a sequence of G -valued randomvariables such that for every j ∈ N ∗ , β j is a product of elements of { α i , α − i , i ∈ N ∗ } : β j = w j (cid:0) ( α i , α − i ) i ∈ N ∗ (cid:1) , with w j being a finite word.Suppose that for any j ∈ N ∗ , there exists an index i j such that α i j appearsexactly once in w j and in no other word ( w j ′ ) j ′ = j . Then ( β j ) ∞ j =1 is a family ofindependent Haar distributed random variables. Proof. Let k be any positive integer and let ( j, j , ..., j k ) be a k + 1-tuple ofpositive integers. Let i j ∈ N ∗ such that α i j appears exactly once in w j and in noother word ( w j ′ ) j ′ = j . Let F : G k → R and f : G → R be two continuous functions.There exist w and w two words in ( α i , α − i ) i = i j , J a subset of N \ { i j } and ˜ F acontinuous function from G J to R such that a.s.: f ( β j ) F ( β j ...β j k ) = f ( w α i j w ) ˜ F (( α i ) i ∈ J ) . Using the translation invariance of the Haar measure: E [ f ( β j ) F ( β j ...β j k )] = E h f ( w α i j w ) ˜ F (( α i ) i ∈ J ) i = Z G E h f ( w xw ) ˜ F (( α i ) i ∈ J ) i dx = Z G E h f ( x ) ˜ F (( α i ) i ∈ J ) i dx = (cid:18)Z G f ( x ) dx (cid:19) E h ˜ F (( α i ) i ∈ J ) i . Thus for any j ∈ N ∗ , β j is a Haar random variable which is independent of( β j , j = i ). (cid:3) In the last subsection, we considered only L´evy processes which were invariantby conjugation by G . Actually for any G -valued self-invariant by conjugation L´evyprocess Y , one can construct a planar Markovian holonomy field associated to Y .Let us recall the notion of support of a process. Definition . Let Y = (cid:0) Y t (cid:1) t ∈ R + be a random process. The support of Y is H Y = (cid:10) S t ∈ R + H Y t (cid:11) . An other formulation is to say that the support of a process, say Y , is thesmallest closed group such that for any t ∈ R + , P ( Y t ∈ H Y ) = 1. If Y is a L´evyprocess, we can consider Y as a process living in H Y . Remark . Let Y be a L´evy process and let us suppose that for any t ≥ e is in Supp ( Y t ). Then for any t > H Y = H Y t . Indeed, using the property that Y is a L´evy process, for any 0 ≤ t < s , Supp ( Y s ) = Supp ( Y t ) Supp ( Y s − t ). Thus,using the condition on the support of Y t , H Y t is increasing in t . Yet, using thesame argument, we see that H Y t = H Y t , thus H Y t does not depend on t > 0. Thisremark explains why we impose that for any t ≥ e ∈ Supp ( Y t ) in the upcomingProposition 8.26.The notion of invariance by conjugation for a random process can be weakened. Definition . A G -valued process ( Y t ) t ≥ is self-invariant by conjugationif it is invariant by conjugation by H Y .Let η be a finite Borel measure on G n . For any g in G , the measure η g on G n is the unique measure such that for any continuous function f : G n → R : η g ( f ) = Z G f ( g − g g, ..., g − g n g ) η ( dg , ..., dg n ) . (7.3)We can now construct a planar Yang-Mills field associated with any self-invariantby conjugation L´evy process. Theorem . For every G -valued self-invariant by conjugation L´evy pro-cess Y , there exists a unique stochastically continuous strong planar Markovianholonomy field (cid:0) E Yvol (cid:1) vol , called the planar Yang-Mills field associated with Y , suchthat for any measure of area vol , for any finite planar graph G , for any rooted span-ning tree T of G and any family of facial loops ( c F ) F ∈ F b oriented anti-clockwise,under E Yvol , the law of (cid:0) h ( l c F ,T ) (cid:1) F ∈ F b is: Z G (cid:0) ⊗ F ∈ F b m vol ( F ) (cid:1) g dg, where ( m t ) t ≥ is the semi-group of convolution of measures associated with Y . Let us notice that on (cid:0) E Yvol , B (cid:1) , (cid:0) h ( l c F ,T ) (cid:1) F ∈ F b is not, in general, a sequence ofindependent variables. Proof. The unicity part uses the same arguments as usual. Let us prove theexistence of the stochastically continuous strong planar Markovian holonomy field (cid:0) E Yvol (cid:1) vol . Let Y = (cid:0) Y t (cid:1) t ≥ be a G -valued self-invariant by conjugation L´evy pro-cess and H Y be the support of Y . Using the discussion after Definition 7.8, wecan see the process Y as a H Y -valued L´evy process which is invariant by conjuga-tion (by H Y ). Thus, applying Propositions 7.3 and 7.5, there exists a H Y -valuedstochastically continuous strong planar Markovian holonomy field such that for anymeasure of area vol , any finite planar graph G , for any rooted spanning tree T of G and any family of facial loops ( c F ) F ∈ F b oriented anti-clockwise, under E Yvol :(1) the random variables ( h ( l c F ,T )) F ∈ F b are independent, .2. CONSTRUCTION OF GENERAL PLANAR YANG-MILLS FIELDS 67 (2) for any F ∈ F b , h ( l c F ,T ) has the same law as Y vol ( F ) .Recall that Proposition 3.15 can also be applied to planar (continuous) Markov-ian holonomy fields. Thus we can extend the group on which E Yvol is defined, from H Y to G : we will denote it E Yvol . It is a G -valued stochastically continuous strongplanar Markovian holonomy field and by definition, for any measure of area vol , forany finite planar graph G , for any rooted spanning tree T of G and any family offacial loops ( c F ) F ∈ F b oriented anti-clockwise, under E Yvol , the law of (cid:0) h ( l c F ,T ) (cid:1) F ∈ F b is: Z G (cid:0) ⊗ F ∈ F b m vol ( F ) (cid:1) g dg. This ends the proof of the theorem. (cid:3) Definition . By construction, the planar Yang-Mills field associated witha self-invariant by conjugation L´evy process Y is constructible. Its restriction tomultiplicative functions on finite planar graphs is called the discrete planar Yang-Mills field associated with Y , we will denote it (cid:0) E Y, G vol (cid:1) G ,vol .We are led to classify the planar Yang-Mills fields according to their degree ofsymmetry. In Section 10, we will prove equivalent conditions in order to classifyplanar Yang-Mills fields. Definition . Let (cid:0) E Yvol (cid:1) vol be a planar Yang-Mills field associated witha G -valued self-invariant by conjugation L´evy process Y = (cid:0) Y t (cid:1) t ∈ R + . The planarYang-Mills field (cid:0) E Yvol (cid:1) vol and the L´evy process Y are pure if (cid:0) Y t (cid:1) t ≥ is invariant byconjugation by G and mixed if not pure. They are also non-degenerate if H Y = G and degenerate if H Y = G . The same definition holds for the discrete planar Yang-Mills field associated with Y .According to this definition, any planar Yang-Mills field is either pure non-degenerate, pure degenerate or mixed degenerate. art 3 Characterization of PlanarMarkovian Holonomy Fields HAPTER 8 Braids and probabilities II: a geometric point ofview, infinite random sequences and randomprocesses In order to characterize planar Markovian holonomy fields, we will use inten-sively the invariance by area-preserving homeomorphisms. The braid group willappear again, as the diffeotopy group of the n -punctured disk which is studied inthe following section. n -punctured disk Let D be the disk of center 0 and radius 1 and Q n = { q k = k − − nn , ≤ k ≤ n } .Let Diff( D , Q n , ∂ D ) be the group of diffeomorphisms of D which fix the set Q n andfix pointwise a neighborhood of ∂ D . The class of isotopy of the identity mappingin Diff( D , Q n , ∂ D ) is a normal subgroup called Diff ( D , Q n , ∂ D ). The diffeotopygroup of the disk with n points is M n ( D ) = Diff( D , Q n , ∂ D ) . Diff ( D , Q n , ∂ D ) . Animportant theorem is that: M n ( D ) ≃ B n . (8.1)This isomorphism is constructed by sending some special elements, the half-twists or Dehn-twists, on the canonical family of generators of the braid groupsdenoted by ( β i ) n − i =1 . A half-twist permutes the points q k and q k +1 for some k anddoes not move the other points ( q i ) i/ ∈{ k,k +1 } . For a precise definition, one considersfor 1 ≤ k ≤ n , t k the isotopy class of the diffeomorphism ˜ t k equals to identityoutside the disk of radius n centered at q k + q k +1 and defined by ˜ t k ( x ) = ψ ◦ t ◦ ψ − ,where: ψ : x n (cid:18) x − q k + q k +1 (cid:19) ,t ( re iθ ) = re i π (cid:0) θ + α ( r ) (cid:1) , and α is a smooth bijection from [0 , 1] to itself, which is equal to 0 on a neighborhoodof 1 and to at .This geometric construction of the braid group allows us to recover the actionof the braid group which was given by Definition 6.2. Indeed, the group M n ( D )acts on the fundamental group of D \ Q n which is isomorphic to F n the free group ofrank n . We will take − i as the base point for the fundamental group of D \ Q n . Let x k be the homotopy class of the loop based at − i which goes only around q k anti-clockwise. One can verify that the action of M n ( D ) on F n , with the identificationgiven in (8.1), is the action given by Definition 6.2. 712 8. BRAIDS AND PROBABILITIES II Figure 1. The braid β k , k = (2 , , G , the fundamental group of G is isomorphic tothe fundamental group of the disk without one point in each of the bounded faces: π ( G ) ≃ π (cid:0)(cid:0) D \ Q | F b | (cid:1)(cid:1) . Thus, we get a natural action of a braid group on thegroup π ( G ) which is isomorphic to the reduced group of loops on G defined in Sec-tion 5.1. A consequence of the existence of such action is the upcoming Proposition9.10. Proposition 6.12 implies that every finite sequence of i.i.d. random variableswhich is auto-invariant by conjugation is invariant by braids. It is natural to wonderif one can characterize finite sequence of random variables which are invariant bybraids. As for exchangeable sequences of random variables, it is easier to work withinfinite sequence of random variables. Definition . An infinite random sequence ξ = (cid:0) ξ i (cid:1) i ∈ N ∗ in G is braidable(or braid-invariant or invariant by braids) if for any integer n greater that 1 andany braid β ∈ B n , the following equality in law holds: β • (cid:0) ξ i (cid:1) ≤ i ≤ n = (cid:0) ξ i (cid:1) ≤ i ≤ n . Let us recall that ξ is spreadable if for any increasing sequence of positiveintegers ( k i ) i ≥ , we have the equality in law: (cid:0) ξ k i (cid:1) ≤ i = (cid:0) ξ i (cid:1) ≤ i . These properties seem to be quite different, yet we are going to prove that onecondition is weaker than the other. Lemma . Any braidable infinite family of random variables is spreadable. Proof. Let k = ( k < k < ... < k n ) be a finite strictly increasing sequenceof integers. Let β k be the braid: β k = β − n ...β − k n − β − ...β − k − β − ...β − k − We have drawn in Figure 1 the braid β k with k = (2 , , i = 1 , ..., n , thelines linking ( i, 1) to ( k i , 0) are behind in the diagram, this braid verifies that for anyelement ( g , ..., g k n ) of G k n , for any integer i between 1 and n , (cid:0) β k • ( g , ..., g k n ) (cid:1) i = g k i . Let ξ = ( ξ i ) i ∈ N ∗ be a braidable random sequence. The following equality inlaw holds β k • (cid:0) ξ i (cid:1) ≤ i ≤ k n = (cid:0) ξ i (cid:1) ≤ i ≤ k n . By restricting it for i between 0 and n − ξ k , ...ξ k n ) = ( ξ , ...ξ n ) , from which one can concludethat ξ is spreadable. (cid:3) Let m be a probability measure on G . We denote by m ⊗∞ the measure on G N ∗ such that the unidimensional projections are independent and identically dis-tributed with law m . Let ξ be an infinite random sequence in G . .2. A DE-FINETTI THEOREM FOR THE BRAID GROUP 73 Definition . Let A be a σ -field. The sequence ξ is i.i.d. conditionally to A if there exists a random measure η on G , A -measurable, such that the conditionaldistribution of ξ given A is η ⊗∞ : P (cid:2) ξ ∈ . | A (cid:3) = η ⊗∞ . It is conditionally i.i.d. ifthere exists a σ -field A such that it is i.i.d. conditionally to A .If ξ is i.i.d. conditionally to A , its law is of the form: Z M ( G ) m ⊗∞ dν ( m ) , where ν is the law of η . If we just want to keep in mind the form of the law of ξ , wewill say that ξ is a mixture of i.i.d. random sequences. Let us state an extensionof de Finetti-Ryll-Nardzewski’s theorem for the braid group. Theorem . The sequence ξ is braidable if and only if it is i.i.d. conditionallyto T ξ = ∩ n ∈ N ∗ σ ( ξ k , k ≥ n ) and conditionally to T ξ , almost surely the law of ξ isinvariant by conjugation by its own support. Proof. An application of Proposition 6.12 to any subsequence of the form( ξ n ) Nn =1 shows that the second condition implies the fact that ξ is braidable.Now, let us suppose that ξ is braidable. As a consequence of Lemma 8.2, theinfinite sequence ξ is spreadable. Using the de Finetti-Ryll-Nardzewski’s theorem(Theorem 1 . Kal05 ]), ξ is i.i.d. conditionally to T ξ . Besides, conditionally to T ξ , ξ is still braidable: an application of Proposition 6.12 shows that conditionallyto T ξ , ( ξ n ) n ∈ N ∗ is an i.i.d. sequence of random variables invariant by conjugationby their own support: the second condition holds. (cid:3) If ξ is braidable, the law of ξ is of the form R M ( G ) m ⊗∞ dν ( m ) , where ν -a.s., m is almost surely invariant by conjugation by its own support. In the next theorem,we give a condition under which one can characterize the mixture which appearsin the last theorem. In order to do so, we consider the diagonal conjugation of G on G N ∗ defined for any g ∈ G and ( x n ) n ∈ N ∗ ∈ G N ∗ by g. ( x n ) n ∈ N ∗ = ( g − x n g ) n ∈ N ∗ .Recall the notion of I -independence defined in Definition 1.31. Definition . The sequence ξ satisfies the property ( P ) if for any integer n ∈ N ∗ , ( ξ k ) k ≤ n and ( ξ k ) k>n are I -independent: for any n, m ≥ 0, any continuousfunctions which are invariant by diagonal conjugation f : G n → R and g : G m → R , f ( ξ , ..., ξ n ) and g ( ξ n +1 , ..., ξ n + m ) are independent. Theorem . Let ξ be an sequence of random variables in G such that: (1) ξ is braidable, (2) it is invariant (in law) by diagonal conjugation: for any g ∈ G , g.ξ hasthe same law as ξ , (3) it satisfies the property ( P ) .There exists m a probability measure on G , invariant by conjugation by its ownsupport, such that the law of ξ is: Z G (cid:0) m ⊗∞ (cid:1) g dg, where (cid:0) m ⊗∞ (cid:1) g is defined using a similar equation as (7.3). Proof. Let ξ be an infinite braidable sequence of G -valued random variableswhich is invariant by diagonal conjugation and satisfies the property ( P ). As a consequence of Theorem 8.4, there exists a random measure η on G , which isalmost surely invariant by conjugation by its own support, such that the condi-tional distribution of ξ given T ξ is η ⊗∞ . Let ν be the law of η , the law of ξ is R M ( G ) m ⊗∞ dν ( m ) . Since ξ is invariant by diagonal conjugation by G , we onlyhave to show that there exists a probability measure m such that the law of ξ onthe invariant σ -field I is equal to m ⊗∞ . Let us remark that this would imply alsothat m is invariant by conjugation by any element of its own support. Let k bea positive integer and f : G k → R be a continuous function invariant by diagonalconjugation: as G k is compact, f is bounded. As the sequence ξ satisfies the prop-erty ( P ), ( f ( ξ ik +1 , ..., ξ ik + k )) i ≥ is an i.i.d. sequence of bounded random variables.Thus, by the law of large numbers, there exists a real l k ( f ) such that:1 n X ≤ i The proof is taken from Serre’s lesson [ SBG79 ] and can be summa-rized in a simple calculation: G ≤ (cid:18) [ g ∈ G g − Hg (cid:19) = (cid:18) [ g ∈ G (cid:0) g − Hg \ e (cid:1) ∪ { e } (cid:19) ≤ G H ( H − 1) + 1 , which can hold if and only if H = G . (cid:3) We can now handle the proof of Proposition 8.7. Proof of Proposition 8.7. It is quite obvious that the assertion 1 impliesthe assertion 2. It remains to prove the other implication. As a consequence ofTheorem 8.6, there exists a probability measure m invariant by conjugation by itsown support such that the law of ξ is R G ( m ⊗∞ ) g dg . Let m be such a probabilitymeasure. As e ∈ Supp ( ξ ), e is in the support of m : the support of m ∗ k isincreasing in k and so is the support of Q ki =1 ξ i . Let k be an integer such that Supp (cid:0) Q ki =1 ξ i (cid:1) = G : for any k ′ ≥ k , Supp (cid:0) Q k ′ i =1 ξ i (cid:1) = G . Let N ≥ k such that Supp ( m ∗ N ) = H m ∗ N . Since N is greater that k , Supp (cid:0) Q Ni =1 ξ i (cid:1) = G . On the otherside, since the law of Q Ni =1 ξ i is R G ( m ∗ N ) g dg , its support is equal to ∪ g ∈ G g − H m ∗ N g .Thus, one has the equality: G = [ g ∈ G g − H m ∗ N g which implies, by Jordan’s theorem (Theorem 8.8), that H m ∗ N = G and then H m = G : the support of m generates the group G . We recall that m was invari-ant by conjugation by its support, hence by H m : m is invariant by conjugationby G and the law of ξ is thus m ⊗∞ . (cid:3) For an arbitrary compact Lie group, it is not true in general that for anymeasure m on G such that e ∈ Supp ( m ), there exists k such that Supp ( m ∗ k ) = H m .Thus, in order to deal with any compact Lie group, we will substitute this fact bythe Itˆo-Kawada’s theorem (Theorem 8.12).As for Jordan’s theorem, it does not hold when G is infinite, as in every compactLie group, any maximal torus intersects all the conjugacy classes. Thus, we haveto impose that the subgroup H intersects every conjugacy class “as much as” G does, which is the meaning of the condition imposed in the upcoming Proposition8.13. Doing so, we will be able to prove the following proposition which holds forany arbitrary compact Lie group. From now on, let G be a compact Lie group. Proposition . Let ξ be an infinite sequence of G -valued random variableswhich is braidable, invariant by diagonal conjugation and satisfies the property ( P ) .Let us suppose that e ∈ Supp ( ξ ) . The following assertions are equivalent: (1) ξ is a sequence of i.i.d. random variables which support generates G . (2) the random variables Q nk =1 ξ k converge in law to a Haar random variableas n goes to infinity. In order to prove this proposition, let us introduce Itˆo-Kawada’s theorem anda measurable version of the theorem of Jordan which holds for any compact Liegroup. Definition . Let m be a probability measure on G . It is: • aperiodic if its support Supp ( m ) is not contained in a left or right propercoset of a proper closed subgroup of G , • non-degenerate if H m = G . Remark . It is obvious that m is non-degenerate if it is seen as a measureon H m . Besides, if e ∈ Supp ( m ) then m is aperiodic.Under the condition of aperiodicity and non-degeneracy, Itˆo-Kawada’s theorem(Theorem 3 . . . of [ Str60 ], first proved in [ KI40 ]) explains the behavior of m ∗ n when n goes to infinity. Theorem . Let µ be a non-degenerate and aperi-odic probability measure on G . The sequence µ ∗ n converges in distribution to thenormalized Haar measure on G as n goes to infinity. Let us state our generalization of Jordan’s theorem, Theorem 8.8, valid for anycompact Lie group. Recall that for any compact Lie group K , we denote by λ K the normalized Haar measure on K . Proposition . Let H be a closed subgroup of G . If Z G λ g − Hg dg = λ G , then G = H . Proof. We want to prove that R G λ g − Hg = λ G : it is enough to construct acontinuous function φ : G → R , invariant by conjugation, such that λ H ( φ ) = λ G ( φ ).The space H \ G of right cosets of H is a nice topological space: it is a differentiablemanifold and there exists ˜ f a non-constant real continuous function on H \ G . Let p : G → H \ G be the canonical projection, the function f = ˜ f ◦ p is a real non-constant square-integrable function f on G invariant by left multiplication by H : .3. DEGENERACY OF THE MIXTURE 77 for any g ∈ G , for any h ∈ H , f ( g ) = f ( hg ) . One can also assume that f is of zeromean on G .Let E = (cid:8) φ ∈ L ( G ) , R G φ ( g ) dλ G ( g ) = 0 (cid:9) be the space of square-integrablezero mean functions on G . The group G acts on E , by left multiplication on theargument and this representation has no non-zero fixed point. On the other hand,the restriction of this representation on H has at least one fixed point, namely f . We can decompose E as a sum of finite dimensional irreducible representationsof G : E = ∞ M i =1 E i . None of the E i is the trivial representation of G as we have restricted the action of G to zero mean functions. The action of H on E admits a fixed point f . We candecompose f on ∞ L i =1 E i . As, for any integer i , the space E i is invariant under theaction of H , there exists at least an integer i such that E i seen as a H -moduleis not irreducible. We denote by χ i the character of the G -module E i . By theclassical theory of character, Z G χ i dλ G = dim( E Gi ) = 0 , whereas: Z G χ i dλ H = Z H χ i dλ H = dim( E Hi ) ≥ , where E Gi and E Hi are the vector spaces of fixed points in E i under the actions of G and H . Thus, we just found a central function χ i such that: Z G χ i dλ G = Z G χ i dλ H . This ends the proof. (cid:3) We have now all the tools in order to prove Proposition 8.9. Proof of Proposition 8.9. As a consequence of Theorem 8.12, Remark 8.11and the fact that e ∈ Supp ( ξ ), it is easy to see that the condition 1 implies condition2. Let us prove the other implication.Let ξ be an infinite sequence of G -valued random variables which is braidable,invariant by diagonal conjugation and which satisfies the property ( P ). Let us sup-pose that e ∈ Supp ( ξ ). As a consequence of Theorem 8.6, there exists a probabilitymeasure m invariant by conjugation by its own support such that the law of ξ is R G ( m ⊗∞ ) g dg . Let m be such a probability measure. Using the hypothesis on ξ , e ∈ Supp ( m ).Let us suppose that Q nk =1 ξ k converges in law to a Haar random variable. Aswe have seen in Remark 8.11, the measure m is aperiodic and non-degenerate ifseen as a measure on H m . Using Itˆo-Kawada’s theorem, when n goes to infinity, m ∗ n converges to the Haar probability measure on H m which we denote by λ H m .For any integer n , the law of n Q i =1 ξ i is R G ( m ∗ n ) g dg and thus, using the hypothesis on the law of n Q i =1 ξ i and our previous discussion, one gets the equality: λ G = Z G λ g − H m g dg. By Proposition 8.13, it follows that H m = G . Since the measure m is invariant byconjugation by H m , it is invariant by conjugation by G : the law of ξ is m ⊗∞ . (cid:3) The following theorem gives weaker conditions on ξ inorder to understand the case when ξ is an infinite sequence of i.i.d. random variablessuch that H ξ = G . We recall that G is a compact Lie group. Theorem . Let ξ be an infinite sequence of G -valued random variableswhich is braidable, invariant by diagonal conjugation and satisfies the property ( P ) .The following assertions are equivalent: (1) the sequence ξ is a sequence of i.i.d. random variables invariant by con-jugation by G , (2) there exists ν a probability measure on G such that for any positive integer n , the law of Q nk =1 ξ k is ν ∗ n . Before proving this theorem, let us recall some basic, yet crucial, results aboutrepresentations and integration. First of all, Peter-Weyl’s theorem asserts that theset of matrix elements of irreducible representations n g v ( π ( g ) w ) , π ∈ ˆ G, v ∈ V ∗ π , w ∈ V π o , where ˆ G is the set of irreducible representations of G , is dense for the uniformnorm in the set of continuous functions on G . Thus, any measure m on G is fullycharacterized by its Fourier coefficients defined as: ∀ π ∈ ˆ G , π ( m ) = Z G π ( g ) m ( dg ) . Secondly, let π : G → Gl ( V ) be an irreducible representation of dimension d π . Let A be a matrix acting on V . By the Schur’s lemma, Z G π ( g ) Aπ ( g ) − dg = T r ( A ) d π Id. Let m be a probability measure on G . Definition . The measure m quasi-invariant by conjugation if there exists ν a probability measure on G such that for any n ∈ N Z G ( m g ) ∗ n dg = ν ∗ n . The main characterization of quasi-invariant by conjugation probability mea-sures is given by the following proposition. Proposition . The measure m is quasi-invariant by conjugation if andonly if for any irreducible representation π of G , the matrix π ( m ) has only oneeigenvalue. .3. DEGENERACY OF THE MIXTURE 79 Proof. Let π be an irreducible representation of G and let n be a positiveinteger. Let us compute π (cid:0)R G ( m ∗ n ) g dg (cid:1) and π (cid:16)(cid:0)R G m g dg (cid:1) ∗ n (cid:17) : π (cid:18)Z G ( m ∗ n ) g dg (cid:19) = Z G π ( gg ′ g − ) m ∗ n ( dg ′ ) dg = Z G π ( g ) (cid:18)Z G π ( g ′ ) m ∗ n ( dg ′ ) (cid:19) π ( g ) − dg = 1 d π T r ( π ( m ) n ) Id,π (cid:18)(cid:18)Z G m g dg (cid:19) ∗ n (cid:19) = π (cid:18)(cid:18)Z G m g dg (cid:19)(cid:19) n = (cid:18) d π T r ( π ( m )) (cid:19) n Id. Proposition 8.16 is equivalent to the following assertion: for any positive inte-ger n , Z g ( m g ) ∗ n dg = (cid:18)Z g m g dg (cid:19) ∗ n (8.2)if and only if for any irreducible representation π of G , the matrix π ( m ) has onlyone eigenvalue. Yet, using the remark about Peter-Weyl’s theorem, Equality (8.2)holds for any positive integer n if and only if for any irreducible representation π ,for any positive integer n : π (cid:18)Z g ( m g ) ∗ n dg (cid:19) = π (cid:18)(cid:18)Z g m g dg (cid:19) ∗ n (cid:19) , hence if and only if for any irreducible representation π , for any positive integer n : T r ( π ( m ) n ) = T r (cid:18)(cid:18) T r ( π ( m )) d π Id (cid:19) n (cid:19) . The proposition is a consequence of the link between the traces of the positivepowers of a finite matrix and the set of its eigenvalues and the fact that the matrix T r ( π ( ν )) d π Id has only one eigenvalue. (cid:3) It is natural to wonder if a quasi-invariant by conjugation probability measure isinvariant by conjugation. The answer is no and we will construct a counter-examplein the symmetric group S . Lemma . Let ( µ t ) t ≥ (reps. ( η t ) t ≥ ) be the continuous semi-group of con-volution of measures starting from δ id on the symmetric group S , associated withthe jump measure m (resp. m ): m ((12)) = 0 , m ((12)) = 1 ,m ((13)) = 1 , m ((13)) = 1 ,m ((23)) = 2 , m ((23)) = 1 ,m ((123)) = 2 , m ((123)) = 1 ,m ((132)) = 0 , m ((132)) = 1 . The measure µ is quasi-invariant by conjugation and for any n ∈ N , Z G ( µ g ) ∗ n dg = η ∗ n . Proof. We have to check that the condition of Proposition 8.16 is fulfilledby µ . Actually we only have to show that for any π ∈ c S , π ( m ) has only oneeigenvalue, which is equal to the one of π ( m ). The group S has only threeirreducible representations two of which have dimension one. It remains to compute π ( m ) where π is the representation of S on { ( a, b, c ) ∈ R , a + b + c = 0 } . We leavethis calculation as an exercise. (cid:3) The quasi-invariance by conjugation property does not imply the invariance byconjugation property, yet, we have the following theorem. Theorem . Let m be a probability measure on G . Let us suppose that m is invariant by conjugation by its own support. Then, m is quasi-invariant byconjugation if and only if m is invariant by conjugation by G . Proof. Almost by definition, any probability measure which is invariant byconjugation is quasi-invariant by conjugation. It remains to prove the “only if”part of the theorem. Let m be a quasi-invariant by conjugation probability mea-sure. Using Proposition 8.16, for any irreducible representation π of G , π ( m ) hasonly one eigenvalue. Any irreducible representation π ∈ ˆ G defines by restrictiona representation of H m . Since H m is a closed subgroup of G , it is a compact Liegroup, thus we can apply Peter-Weyl’s theorem which allows us to decompose anyrepresentation of H m as a direct sum of irreducible representations: π = n M i =1 π i , with π i ∈ ˆ H m . Since m is invariant by conjugation by H m , thanks to Schur’slemma, for any i ∈ { , ..., n } , π i ( m ) is a scalar matrix, hence π ( m ) is diagonal. Asit has only one eigenvalue, it is a multiple of the identity. Let π ∈ ˆ G acting on V ,let w ∈ V , v ∈ V ∗ and let h ∈ G : Z G v ( π ( hgh − ) w ) m ( dg ) = v (cid:18) π ( h ) (cid:18)Z G π ( g ) m ( dg ) (cid:19) π ( h ) − w (cid:19) = v (cid:18)(cid:18)Z G π ( g ) m ( dg ) (cid:19) w (cid:19) = Z G v ( π ( g ) w ) m ( dg ) . Thus, using Peter-Weyl’s theorem, m is invariant by conjugation by G . (cid:3) We have now all the tools in order to prove Theorem 8.14. Proof of Theorem 8.14. Let ξ be an infinite sequence of G -valued randomvariables which is braidable, invariant by diagonal conjugation and which satisfiesthe property ( P ). As a consequence of Theorem 8.6, there exists a probabilitymeasure m invariant by conjugation by its own support such that the law of ξ is R G ( m ⊗∞ ) g dg . .4. PROCESSES AND THE BRAID GROUP 81 Let us suppose that ξ is a sequence of i.i.d. random variables, then one cantake as m the law of ξ : the law of ξ is equal to m ⊗∞ and thus, for any n , the lawof Q nk =1 ξ k is m ∗ n .Instead of assuming that ξ is a sequence of i.i.d. random variables, let ussuppose that there exists ν a probability measure on G such that for any n , thelaw of Q nk =1 ξ k is ν ∗ n . The law of Q nk =1 ξ k is R G ( m ∗ k ) g dg : it shows that theprobability measure m is quasi-invariant by conjugation. Yet, it is also invariantby conjugation by its own support. By Theorem 8.18, m is invariant by conjugationby G and thus ξ is a sequence of i.i.d. random variables. (cid:3) Sections 8.2 and 8.3 can be generalized in order to get similar results for G -valued processes indexed by R + . We define the rational increments of a process as Kallem-berg does in [ Kal05 ]. Let ( X t ) t ∈ R + be a G -valued random process indexed by R + . Definition . We define the (rational) increments of ( X t ) t ∈ R + for n in N ∗ ∪ ( N ∗ ) − and j ≥ X n,j = X − j − n X jn . (8.3)The process ( X t ) t ∈ R + has spreadable (resp. braidable) increments if for every n ∈ N ∗ ∪ ( N ∗ ) − , the sequence ( X n,i )
We will show that 1 implies 2, 3 implies 1 and at last 2 implies 3.Let us assume that X has auto-invariant by conjugation increments. Let t ∈ R + , then ( X t , X − t X t ) is auto-invariant by conjugation: X t is invariant byconjugation by the support of X − t X t . As X is a L´evy process, X − t X t and X t has the same law, thus the same support: X t is invariant by conjugation by its ownsupport. Now, let us show that 3 implies 1. Let us assume that X is invariant byconjugation by H X . By definition for any t ∈ R + , Supp ( X t ) ⊂ H X . Let t < t and t < t be four non negative reals. As the process X is invariant by conjugationby H X , X − t X t is invariant by Supp ( X t − t ) and thus by Supp ( X − t X t ). Thisimplies easily that X has auto-invariant by conjugation increments.It remains to prove that 2 implies 3. Let us assume that for any t ∈ R + , X t isinvariant by conjugation by its own support. Let us first remark that, if U and V aretwo random independent variables in G , Supp ( U V ) = Supp ( U ) . Supp ( V ) . Besides,if they are both invariant in law by conjugation by a set S , U V is also invariantby conjugation by S . Moreover, if U is invariant by conjugation by any element of S , it is invariant by conjugation by any element of the closure of the semi-groupgenerated by S : S ∞ k =1 S k , which, in the case where G is compact, is a group. Let n be a positive integer and let t be a positive real. Using the hypothesis on X , X tn is invariant by conjugation by Supp ( X tn ). Taking n independent copies of X tn andapplying the remarks above, we find that X t is still invariant by conjugation by Supp ( X tn ) and thus also for any integer k ≥ 1, by Supp ( X tn ) k = Supp ( X kn t ), or bythe semi-group generated by Supp ( X kn t ), which is nothing but H X kn t . Thus, X t isinvariant by conjugation by: [ q ∈ Q + H X qt . Since the laws of (cid:0) X t (cid:1) t ≥ form a continuous semi-group of convolution of mea-sures, for any s ≥ X s ∈ S q ∈ Q + H X q.t a.s. and thus H X s ⊂ S q ∈ Q + H X qt , hence theequality: H X = [ q ∈ Q + H X qt . Thus, for any positive real t , X t is invariant by conjugation by H X . Using the factthat X has independent and stationary increments, it implies that the L´evy process X is self-invariant by conjugation. (cid:3) We can now state the general-ization of B¨uhlmann’s theorem (Theorem 1.19 of [ Kal05 ]) for the braid group. Theorem . Let X be a G -valued stochastically continuous process indexedby R + with X = e . The following conditions are equivalent: (1) X has braidable increments, (2) X is a mixture of self-invariant by conjugation L´evy processes.The σ -field which makes the rational increments, as defined in Definition 8.19,conditionally i.i.d. is the σ -field T = ∩ t ∈ Q + σ ( X − t X s , s > t ) . Besides, the followingconditions are equivalent: (1) X is invariant by conjugation by G and has braidable and I -independentincrements, (2) there exists a self-invariant by conjugation L´evy process Y , such that thelaw of X is U Y U − , where U is a Haar variable on G independent of Y . Proof. Let us consider the first part of the theorem. Let us show that thecondition 2 implies the first one: it is enough to show that any self-invariant byconjugation L´evy process has braidable increments. Let Z be a self-invariant by .4. PROCESSES AND THE BRAID GROUP 83 conjugation L´evy process. By Proposition 8.22, for any n ∈ N ∗ ∪ ( N ∗ ) − , thesequence of increments ( Z n,j ) j , defined in Definition 8.19, is a sequence of i.i.d.random variables which are invariant by conjugation by their own support. Hence,by Theorem 8.4, it is braidable: the process Z has braidable increments.Now, let us consider X a G -valued stochastically continuous process indexedby R + with X = e . Let us suppose that X has braidable increments. Followingthe proof of Theorem 1 . 19 of [ Kal05 ], we introduce the processes: Y kn ( t ) = X (cid:18) k − n (cid:19) − X (cid:18) t + k − n (cid:19) , t ∈ Q ∩ [0 , n − ] , k ∈ N ∗ , n ∈ N ∗ Let n be a positive integer. Using the same arguments used in Lemma 8.2, no-tice that the sequence ( Y kn ) k ∈ N ∗ is spreadable. Applying to these sequences thedeFinetti-Ryll-Nardzewski’s theorem (Theorem 1 . . Kal05 ])which is valid for sequences in Polish spaces, we conclude that for any n ∈ N ∗ ,conditionally to the tail σ -field T n , the sequence ( Y kn ) k ∈ N ∗ is a sequence of i.i.d.random variables. We considered t ∈ Q + ∩ [0 , n − ] in the definition of Y kn as theproduct of a countable family of Polish spaces is still a Polish space. The σ -field T n we are conditioning on is a.s. independent of n : we call it T . Given T , X hasconditionally stationary independent (rational) increments. For any t ∈ Q + , let m t be the law of X t conditionally to T : for any t ∈ Q + and any s ∈ Q + , almost surely m t ∗ m s = m t + s . Besides, using the stochastic continuity of X and the fact that X = e , one has that almost surely ( m t ) t ∈ Q + is uniformly continuous. We can ex-tend the semi-group ( m t ) t ∈ Q + in order to get a semi-group ( m t ) t ∈ R + : by stochasticcontinuity the process X is then a mixture of L´evy processes. Let us consider apositive rational number q . Using a similar argument as in the proof of Theorem8.4, applied to the sequence ( X nq ) n ∈ N , conditionally on T , the random variable X q is invariant by conjugation by its own support. Using a continuity argument allowsus to extend the result for any q ∈ R + . The result follows from Proposition 8.22: X is a mixture of self-invariant by conjugation L´evy process.The second part of the theorem is deduced from Theorem 8.6 applied to theincrements of X . (cid:3) In this subsection, Sections 8.3.1 and8.3.2 are generalized in the setting of processes. The proofs will be omitted sincethe results follow directly from their counterpart in the setting of sequences andfrom Theorem 8.23. Recall Definition 7.13, where the notions of pure/mixed,degenerate/non-degenerate L´evy processes were defined. Let us state the conse-quence of Proposition 8.7. Proposition . Let G be a finite group. Let X be a G -valued stochasticallycontinuous process invariant by conjugation by G such that X = e and which hasbraidable and I -independent increments. The following assertions are equivalent: (1) X is a pure non-degenerate L´evy process, (2) there exists t ∈ R + such that Supp ( X t ) = G .If one of the two conditions holds then for any t ∈ R + , Supp ( X t ) = G . Let us remark that, in order to prove the last proposition, we have to replacethe property of Supp ( m ∗ k ) we used in the proof of Proposition 8.7 by the followingstraightforward lemma. Lemma . Let G be a finite group and let (cid:0) Y t (cid:1) t ≥ be a L´evy process on G .For any real t ≥ , Supp ( Y t ) = H Y . Let us state the consequence of Proposition 8.9. From now on, let G be acompact Lie group. Proposition . Let X be a G -valued stochastically continuous process in-variant by conjugation by G such that X = e and which has braidable and I -independent increments. Let us suppose that e is in Supp ( X t ) for any t ∈ R + . Thefollowing conditions are equivalent: (1) the process X is a pure non-degenerate L´evy process, (2) the random variable X t converges in law to a Haar random variable on G when t goes to infinity. In order to conclude this section, it remains to state the consequence of Theo-rem 8.14. Theorem . Let X be a G -valued stochastically continuous process invariantby conjugation by G such that X = e and which has braidable and I -independentincrements. The following assertions are equivalent: (1) the process X is a pure L´evy process, (2) there exists a G -valued L´evy process Z such that for any t ∈ R + , X t hasthe same law as Z t . Remark . If the increments of X are exchangeable then the theorem is nomore valid. Let us consider the L´evy process Y (respectively Z ) associated with( µ t ) t ≥ (respectively ( η t ) t ≥ ) defined in Lemma 8.17. Let U be a random Haarvariable which is independent from Y and Z and let X = U Y U − . The process X is stochastically continuous, invariant by conjugation by G , has exchangeable and I -independent increments and for any t ≥ X t has the same law as Z t . Yet, theprocess X is not a pure Lvy process since Y is not invariant by conjugation by G .HAPTER 9 Characterization of Stochastically Continuous inLaw Weak Discrete Planar Markovian HolonomyFields The characterization of stochastically continuous in law weak discrete planarMarkovian holonomy fields is given by the following theorem whose proof will bethe main goal of this section. Theorem . Let (cid:0) E G vol (cid:1) G ,vol be a G -valued stochastically continuous in lawweak discrete planar Markovian holonomy field. There exists a G -valued L´evy pro-cess, ( Y t ) t ≥ , self-invariant by conjugation such that (cid:0) E G vol (cid:1) G ,vol is the weak discreteplanar Yang-Mills field associated with ( Y t ) t ≥ . This means that for any measure ofarea vol and any graph G ∈ G (cid:0) Aff ( R ) (cid:1) , E G vol is equal to E Y, G vol , where (cid:16) E Y, G vol (cid:17) G ,vol is the discrete planar Yang-Mills field associated with ( Y t ) t ≥ .If G is Abelian, the L´evy process is unique and is characterized by the fact thatfor any simple loop l in Aff ( R ) , under E G ( l ) vol (see Example 2.3), h ( l ) has the samelaw as Y vol ( Int ( l )) . Remark . If G is a non Abelian group, the L´evy process ( Y t ) t ≥ is notunique: it is unique up to an equivalence. Two L´evy processes are equivalent ifthey have the same law when we restrict their law to the invariant σ -field on G R + .Theorem 9.1 asserts that there exists a one-to-one correspondence between the setof equivalence classes of G -valued self-invariant by conjugation L´evy processes andthe set of G -valued stochastically continuous in law weak discrete planar Markovianholonomy fields. In Section 9.1.1, we show that the two-dimensional time objects which arethe planar Markovian holonomy fields are characterized by a one-dimensional timeprocess. In Section 9.1.2, it is shown that this one-dimensional time process has I -independent increments: this allows us to prove Theorem 9.1 when the group G isabelian. In general, the result follows from the braidability of the one-dimensionaltime process which is proved in Section 9.1.3. We can go further than Proposi-tion 5.15 when one considers stochastically continuous in law weak discrete planarMarkovian holonomy fields. Definition . For any 0 ≤ s ≤ t we define ∂c ts = ( s, → ( t, → ( t, → ( s, → ( s, , and p s = (0 , → ( s, 856 9. CHARACTERIZATION OF PLANAR MARKOVIAN HOLONOMY FIELDS the same finite planar graph. Recall the notion of reduced loops defined in thebeginning of Section 5.1. The reduced loop L ts is: L ts = [ p s ∂c ts p − s ] ≃ , . Remark . These loops satisfy the following equalities: L tr = L ts L sr , ∀ ≤ r ≤ s ≤ t,L i +1 i = L i, , ∀ i ∈ N , where we considered the reduced product in the first equality and where the family( L i,j ) i,j was defined in Definition 5.9.The following lemma is a straightforward application of Theorem 7.11. Lemma . Let Y be a G -valued self-invariant by conjugation L´evy process.Let U be a Haar variable on G which is independent from Y . Let (cid:0) E Yvol (cid:1) vol be theplanar Yang-Mills field associated with Y . Under the measure E Ydx , ( h ( L t )) t ∈ R + hasthe same law as (cid:0) U Y t U − (cid:1) t ∈ R + . The following theorem asserts that the process ( h ( L t )) t ∈ R + characterizes anystochastically continuous in law weak discrete planar Markovian holonomy field. Theorem . Let (cid:0) E G vol (cid:1) G ,vol and (cid:16) ˜ E G vol (cid:17) G ,vol be two stochastically continu-ous in law weak discrete planar Markovian holonomy fields. The three followingassertions are equivalent: (1) (cid:0) E G vol (cid:1) G ,vol and (cid:16) ˜ E G vol (cid:17) G ,vol are equal, (2) ( h ( L t )) t ∈ R + has the same law under E Aff dx as under ˜ E Aff dx , (3) for any positive real α , ( h ( L n, )) n ∈ N has the same law under E N αdx asunder ˜ E N αdx . We remind the reader that E Aff dx and E N dx were defined in Remark 3.6. Theproof will consist in proving the equivalence between the conditions 1 and 2, thenbetween 2 and 3. Proof. Since condition 1 clearly implies condition 2, let us show that condition2 implies condition 1. Let us suppose that ( h ( L t )) t ∈ R + has the same law under E Aff dx as under ˜ E Aff dx . Let vol be a measure of area and let G be a finite planargraph in G (cid:0) Aff (cid:0) R (cid:1)(cid:1) . We have to show that E G vol = ˜ E G vol . The proof will consistin a sequence of simplifications, changing the graph and the measure of area littleby little. By Proposition 1.37, these measures are characterized by the way theyintegrate functions of the form: h f ( h ( l ) , ..., h ( l m )), where f is a continuousfunction invariant by conjugation and l , ..., l n are elements of L v ( G ), where v is anygiven vertex of G . Thus, we have to show that (cid:0) E G vol (cid:1) | L v ( G ) = (cid:16) ˜ E G vol (cid:17) | L v ( G ) . Sincethese two measures are defined on multiplicative functions on loops, we can supposethat G is simple. Let us consider a sequence of generic and simple graphs ( G n ) n ≥ which approximate the graph G in the sense of Lemma 2.29. Using the stochasticcontinuity in law of (cid:0) E G vol (cid:1) G ,vol (cid:0) resp. (cid:16) ˜ E G vol (cid:17) G ,vol (cid:1) , the measure (cid:0) E G vol (cid:1) | L v ( G ) (cid:0) resp. .1. PROOF OF THE CHARACTERIZATION THEOREM 87 F F F F F F F ' F F' F F' F F'F F' Figure 1. Graphs G and ˜ G . Figure 2. Graphs G and G . (cid:16) ˜ E G vol (cid:17) | L v ( G ) (cid:1) can be recovered using the sequence of measures (cid:18)(cid:16) E G n vol (cid:17) | L v ( G n ) (cid:19) n ∈ N (cid:0) resp. (cid:18)(cid:16) ˜ E G n vol (cid:17) | L v ( G n ) (cid:19) n ∈ N (cid:1) . This allows us to suppose that G is also generic.Using Corollary 2.31, there exists G ′ a subgraph of the N graph such that theset of G − G ′ piecewise diffeomorphisms is not empty: let ψ be such a homeomor-phism. Let vol ′ be a measure of area on R such that for any bounded face F of G , vol ′ ( ψ ( F )) = vol ( F ). Using the Axiom wDP , we know that E G ′ vol ′ ◦ ψ − = E G vol . By definition of E N vol ′ , the measure E G ′ vol ′ is equal to (cid:16) E N vol ′ (cid:17) |M ult ( P ( G ′ ) ,G ) . The samediscussion holds for (cid:16) ˜ E G vol (cid:17) G ,vol . Thus, if we show that, for any measure of area vol ′ , E N vol ′ = ˜ E N vol ′ , we will get that (cid:0) E G vol (cid:1) G ,vol and (cid:16) ˜ E G vol (cid:17) G ,vol are equal.Let vol ′ be a measure of area on R . Since { L i,j , i, j ∈ N } is a family whichgenerates RL ( N ) and since we are considering gauge-invariant measures, we onlyhave to prove that ( h ( L i,j )) i,j has the same law under E N vol ′ as under ˜ E N vol ′ . Let usshow that it is actually enough to know that ( h ( L n, )) n ∈ N has the same law under E N vol ′ as under ˜ E N vol ′ .Let us consider the two finite planar graphs G and ˜ G drawn in Figure 1. Let vol ′′ be a measure of area such that vol ′′| F ∞ = vol ′| ˜ F ∞ , where F ∞ (resp. ˜ F ∞ ) is theunbounded face of G (resp. ˜ G ). Besides, we impose that the following conditionholds for vol ′′ : ∀ i ∈ { , ..., } , vol ′′ ( F ′ i ) = vol ′ ( F i ) . (9.1)The loops ( L i,j ) i,j ∈{ , } belong to L (0 , ( G ) and the loops ( L i, ) i ∈ [ | , | ] are in L (0 , ( ˜ G ). Let us approximate the loops ( L i,j ) i,j ∈{ , } by loops whose intersectionis reduced to the base point. Such loops are drawn in bold in the left part of Figure 2. The two graphs G and G drawn in Figure 2 satisfy the hypothesis ofTheorem 2.15 and are in G (cid:0) Aff ( R ) (cid:1) : they are homeomorphic. Thus, by Proposi-tion 2.25, there exists an orientation-preserving G − G piecewise diffeomorphismwhich we denote by ψ . We can suppose, up to a modification of G and G whichwill not change the general form of both graphs and thus will not invalidate thediscussion, that vol ′ ( F ) = vol ′′ ( ψ ( F )) for any bounded face F of G . This lastassertion is essentially due to the condition (9.1) on vol ′′ . Using Axiom wDP and using the stochastic continuity in law property, we conclude that under E N vol ′ (resp. ˜ E N vol ′ ), ( h ( L , ) , h ( L , ) , h ( L , ) , h ( L , )) has the same law as ( h ( L i, )) i ∈ [ | , | ] under E N vol ′′ (resp. ˜ E N vol ′′ ). A slight generalization of these arguments allows us toshow that for any integer n there exists a measure of area vol ′′ such that under E N vol (resp. ˜ E N vol ), ( h ( L ,n − ) , ..., h ( L , ) , ..., h ( L n − ,n − ) , ..., h ( L n − , )) has the same lawas ( h ( L i, )) i ∈ [ | ,n − | ] under E N vol ′′ (resp. ˜ E N vol ′′ ).Thus it is now enough to show that for any measure of area vol ′′ , ( h ( L i, )) i ∈ N has the same law under E N vol ′′ as under ˜ E N vol ′′ . Let n ∈ N , let S be the graphdefined as the intersection of the N graph and [0 , n + 1] × [0 , ψ such that its restriction on R + × R isgiven by: ψ : R + × R → R ( x, y ) ( vol ′′ ([0 , x ] × [0 , , y ) . The image of S by ψ , ψ ( S ) is a simple graph in G ( Aff ( R )) and for any bounded face F of S , vol ′′ ( F ) = dx ( ψ ( F )). Let us define for any i ∈ { , ..., n + 1 } , t i = vol ′′ ( L i ).We can apply the Axiom wDP to the two graphs S and ψ ( S ), to the two measuresof area vol ′′ and dx and to ψ . It shows that under E N vol ′′ (resp. ˜ E N vol ′′ ), ( h ( L i, )) ni =0 has the same law as ( h ( L t i +1 t i )) ni =0 under E Aff dx (resp. ˜ E Aff dx ). It remains to show thatfor any integer n , any sequence of positive reals t < ... < t n , ( h ( L t i +1 t i )) ni =0 has thesame law under E Aff dx as under ˜ E Aff dx . Yet, we started with the fact that ( h ( L t )) t ∈ R + has the same law under E Aff dx as under ˜ E Aff dx . This allows us to conclude.Let us prove the equivalence between the conditions 2 . and 3 . Suppose that forany positive real α , ( h ( L n, )) n ∈ N has the same law under E N αdx as under ˜ E N αdx . Bythe Axiom wDP , we can change E N αdx (resp. ˜ E N αdx ) by E Aff αdx (resp. ˜ E Aff αdx ). As anapplication of the Axiom wDP , with ψ given by ψ : ( x, y ) ( αx, y ) , the randomvector (cid:16) h ( L α ( n +1) α.n ) (cid:17) n ∈ N has the same law under E Aff dx as under ˜ E Aff dx . Using the factthat for any positive integers p and q : h (cid:18) L pq (cid:19) = p − Q i =0 h (cid:18) L i +1 qiq (cid:19) , we can concludethat ( h ( L t )) t ∈ Q + has the same law under E Aff dx as under ˜ E Aff dx and by stochasticallycontinuity the same assertion holds for ( h ( L t )) t ∈ R + . Thus, condition 3 . impliescondition 2. The other implication can be proved using the same arguments. (cid:3) Let (cid:0) E G vol (cid:1) G ,vol be a stochastically continuous in law weak discrete planar Markovian holonomy fieldand E Aff dx be the usual expectation associated with it. Definition . Until the end of this section, we set, for any 0 ≤ s ≤ t , Z ts = h ( L ts ) and Z t = Z t . .1. PROOF OF THE CHARACTERIZATION THEOREM 89 Remark . Using the multiplicativity property of random holonomy fieldsand Remark 9.4, for any 0 ≤ r ≤ s ≤ t , Z tr = Z sr Z ts , hence for any 0 ≤ s ≤ t , Z ts = ( Z s ) − Z t . Since Z ts = h (cid:0) p s ∂c ts p − s (cid:1) , by Remark 1.38, under E Aff dx , Z ts has the same law as h ( ∂c ts ). Besides the left translation by s sends dx on itself and ∂c ts on ∂c t − s :applying Axiom wDP (Definition 3.4), we get that under E Aff dx , Z ts has the same law as Z t − s . Moreover, using the stochastic continuity property, under E Aff dx , the process ( Z t ) t ≥ is stochastically continuous and Z is equal to the neutral element of G .A simple but important lemma is the following. Lemma . Under E Aff dx , for any t > , for any finite subset T of [0 , t ] andany finite subset T ′ of [ t , ∞ [ , ( Z t ) t ∈ T and ( Z − t Z t ) t ∈ T ′ are I -independent. Thismeans that for any continuous functions f : G T → R and f ′ : G T ′ → R invariantby diagonal conjugation by G , E Aff dx h f (cid:0) ( Z t ) t ∈ T (cid:1) f ′ (cid:16)(cid:0) Z − t Z t (cid:1) t ∈ T ′ (cid:17)i = E Aff dx (cid:2) f (cid:0) ( Z t ) t ∈ T (cid:1)(cid:3) E Aff dx h f ′ (cid:16)(cid:0) Z − t Z t (cid:1) t ∈ T ′ (cid:17)i . Proof. Let t > T be a finite subset of [0 , t ] and T ′ be a finite subset of[ t , ∞ [. Obviously we can suppose that T ′ ⊂ ] t , ∞ [. Let t ′ be any real strictlygreater that t such that T ′ ⊂ [ t ′ , ∞ [. We remind the reader that for any t ∈ T ′ , Z − t ′ Z t = h (cid:0) p t ′ ∂c tt ′ p − t ′ (cid:1) , thus for any continuous functions f : G T → R and f ′ : G T ′ → R invariant bydiagonal conjugation by G : E Aff dx h f (cid:0) ( Z t ) t ∈ T (cid:1) f ′ (cid:18)(cid:16) Z − t ′ Z t (cid:17) t ∈ T ′ (cid:19) i = E Aff dx h f (cid:0) ( h ( L t )) t ∈ T (cid:1) f ′ (cid:0) ( h ( ∂c tt ′ )) t ∈ T ′ (cid:1)i . Let us denote by t the maximum of T ′ . The loop L t is in Int ( L t ) for any t ∈ T and the loop ∂c tt ′ is in Int ( ∂c t t ′ ) for any t ∈ T ′ . Besides Int ( L t ) ∩ Int ( ∂c t t ′ ) = ∅ .Using the Axiom wDP : E Aff dx (cid:20) f (cid:0) ( Z t ) t ∈ T (cid:1) f ′ (cid:18)(cid:16) Z − t ′ Z t (cid:17) t ∈ T ′ (cid:19)(cid:21) = E Aff dx (cid:2) f (cid:0) ( Z t ) t ∈ T (cid:1)(cid:3) E Aff dx (cid:20) f ′ (cid:18)(cid:16) Z − t ′ Z t (cid:17) t ∈ T ′ (cid:19)(cid:21) . The stochastic continuity of E Aff dx and taking the limit t ′ → t allows us to concludethe proof. (cid:3) When G is Abelian, for any n -tuple ( g , ..., g n ) of elements of G , the diagonalconjugacy class of ( g , ..., g n ) is reduced to (cid:8) ( g , ..., g n ) (cid:9) . Thus, the last lemmaasserts that σ (cid:0) { Z t , t ≤ t } (cid:1) is independent of σ (cid:0) { Z − t Z t , t ≥ t } (cid:1) . Using Remark9.8, this implies that ( Z t ) t ∈ R + is a L´evy process. Applying Theorem 9.6 and Lemma9.5, we deduce that (cid:0) E G vol (cid:1) G ,vol is the planar Yang-Mills field associated with theL´evy process ( Z t ) t ≥ . The Abelian part of Theorem 9.1 is thus proved.When G is not Abelian, we have to get rid of the conjugacy classes in Lemma9.9: it is what we intend to do in the following subsection. FF FFF FF FF F'' ' '' Figure 3. The graphs G and G . Recall the notions andnotations set in Definition 8.19. Proposition . Under E Aff dx , the process ( Z t ) t ∈ R + has braidable increments. Proof. The proof will be essentially graphical. The braid group with m strands is generated by the elementary braids ( β i ) m − i =1 defined in Section 6. This al-lows us to reduce the braidability condition to the fact that for any n ∈ N ∗ ∪ ( N ∗ ) − ,any positive integers m and i such that i < m , the following equality in law holds: β i • (cid:0) Z n, , ..., Z n,m (cid:1) = (cid:0) Z n, , ..., Z n,m (cid:1) . The proof does not depend on the value of n , we will suppose it is equal to 1. Weremind the reader that Z ,i = h (cid:0) p i − ∂c ii − p − i − (cid:1) : we have to understand the lawof the random variables associated with m lassos. Using the stochastic continuityof E Aff dx , we can “shrink” the meander of these lassos and we can suppose that theirintersection is reduced to the base point as we did in the proof of Theorem 9.6.Let i be a positive integer, we will focus only on what happens in the interior of ∂c i +1 i − . Let us consider the graphs G and G drawn in Figure 3. They representwhat is happening in ∂c i +1 i − : the loops in bold represent the part of the i th and i + 1 th lassos inside ∂c i +1 i − and we added to it two paths in dots in order to considersimple graphs. The two graphs satisfy the hypothesis of Theorem 2.15 thus, theyare homeomorphic. Let us consider an orientation-preserving homeomorphism φ between G and G which sends F i on F ′ i for any i ∈ { , ..., } . By Proposition 2.25,there exists an orientation-preserving G − G piecewise diffeomorphism ψ which isequivalent to φ on G : it sends F i on F ′ i for any i ∈ { , ..., } . It is possible to takeit such that ψ is the identity on the unbounded face of G . Besides, one can remarkthat G is the horizontal flip of G : for any integer i ∈ { , ..., } , dx ( F i ) = dx ( F ′ i ).Thus for any bounded face F of G , dx ( ψ ( F )) = dx ( F ). Using the area-preservinghomeomorphism invariance, namely Axiom wDP , E G dx ◦ ψ − = E G dx . Lettingthe shrinking parameter to zero in this equality allows us to recover the followingequality in law: under E Aff dx , β i • (cid:0) Z , , ..., Z ,m (cid:1) = (cid:0) Z , , ..., Z ,m (cid:1) . (cid:3) Recall that we are working under E Aff dx . Using the results of Section 9.1.2 and9.1.3, we already know that the process Z = ( Z t ) t ∈ R + is invariant by conjugationby G and has braidable and I -independent increments. By Theorem 8.23, thereexists a self-invariant by conjugation L´evy process Y , such that the law of Z is U Y U − , where U is a Haar variable on G independent of Y . Lemma 9.5, combinedwith Theorem 9.6 allows us to finish the proof of Theorem 9.1. .2. CONSEQUENCES OF THE CHARACTERIZATION THEOREM 91 Theorem . For a discrete planar Markovian holonomy field, the followingconditions are equivalent: • it is stochastically continuous in law, • it is regular.If the discrete planar Markovian holonomy field is a weak one, then one can replacethe regularity condition by the locally stochastically -H¨older continuity. Proof. We already saw in Corollary 3.13 that, depending if we are workingwith weak or strong discrete planar Markovian holonomy fields, the regularity orthe locally stochastically -H¨older continuity implies the stochastically continuityin law of the discrete planar Markovian holonomy field.Besides if a discrete planar Markovian holonomy field (cid:0) E G vol (cid:1) G ,vol is stochasti-cally continuous in law, its restriction to the piecewise affine graphs is a stochasti-cally continuous in law weak discrete planar Markovian holonomy field. By Theo-rem 9.1, there exists a planar Yang-Mills field (cid:0) E Yvol (cid:1) vol such that (cid:0) E G vol (cid:1) G ∈G ( Aff ( R )) ,vol = (cid:0) ( E Yvol ) |M ult ( P ( G ) ,G ) (cid:1) G ∈G ( Aff ( R )) ,vol . Using the stochastic continuity in law of both of the fields, this equality holds with-out the restriction on the graphs. Using the proof of Proposition 7.3, up to a slightmodification since Y is only self-invariant by conjugation, (cid:0) ( E Yvol ) |M ult ( P ( G ) ,G ) (cid:1) G ,vol is locally stochastically -H¨older continuous and continuously area-dependent, thus (cid:0) E G vol (cid:1) G ,vol is also locally stochastically -H¨older continuous and continuously area-dependent. (cid:3) In Section 3 we have defined four different notions of planar Markovian ho-lonomy fields. By now, we know that, by restriction, a strong planar Markovianholonomy field defines a weak continuous one. Using the results of Section 4, aweak planar Markovian holonomy field defines, when restricted, a weak discreteplanar Markovian holonomy field. Theorem 9.1 now allows us to show that thefour different notions are in some sense equivalent when one considers stochasticallycontinuous objects. Indeed, a stochastically continuous in law weak discrete planarMarkovian holonomy field is the restriction of a planar Yang-Mills field, which bythe results of Section 7 was shown to be a stochastically continuous strong planarMarkovian holonomy field. Besides, by construction, any planar Yang-Mills field isconstructible. This discussion allows us to state the following theorems. Theorem . Let ( E vol ) vol be a family of stochastically continuous randomholonomy fields. We have equivalence between: (1) ( E vol ) vol is a strong planar Markovian holonomy field, (2) (cid:16) ( E vol ) |M ult ( Aff ( R ) ,G ) (cid:17) vol is a weak planar Markovian holonomy field, (3) (cid:16) ( E vol ) |M ult ( P ( G ) ,G ) (cid:17) G ,vol is a strong discrete planar Markovian holonomyfield, (4) (cid:16) ( E vol ) |M ult ( P ( G ) ,G ) (cid:17) G ∈G ( Aff ( R )) ,vol is a weak discrete planar Markovianholonomy field, (5) ( E vol ) vol is a planar Yang-Mills field associated with a L´evy process whichis self-invariant by conjugation. Thus, any stochastically continuous strong planar Markovian holonomy field is con-structible. Theorem . Any G -valued stochastically continuous in law weak discreteplanar Markovian holonomy field is the restriction of a unique G -valued stochasti-cally continuous strong planar Markovian holonomy field. We encourage the reader to have a look at the diagram page 118 where we drawnthe different links between all the notions introduced or used in this paper. The lastconsequence of Theorem 9.1 is the Proposition 3.16. Before giving the proof of thisproposition, we will construct an explicit G -valued stochastically continuous in lawdiscrete planar Markovian holonomy field ( E G vol ) G ,vol which satisfies the hypothesisof Proposition 3.16 but for which the natural restriction defined by the Equation(3.3) is not a discrete planar Markovian holonomy field.For this, we consider the symmetrical group G = S . Let H be the subgroupof G isomorphic to Z / Z which contains the neutral element e and the two 3-cycles c and c . Let X be a H -valued L´evy process which jumps only by multiplicationby c . As H is abelian, X is a self-invariant by conjugation G -valued L´evy processand, because of the condition on the jumps, for any positive time t : P [ X t = c ] = P (cid:2) X t = c (cid:3) . (9.2)Let (cid:0) E Xvol (cid:1) vol be the G -valued planar Yang-Mills field associated with X and let usconsider the restriction (cid:0) E X, G vol (cid:1) G ,vol of (cid:0) E Xvol (cid:1) vol to the finite planar graphs: it is a G -valued stochastically continuous in law discrete planar Markovian holonomy field.Since H is normal in G , (cid:0) E X, G vol (cid:1) G ,vol satisfies the conditions stated in Proposition3.16. The natural restriction of (cid:0) E X, G vol (cid:1) G ,vol , as defined by the Equation (3.3), anddenoted by (cid:0) ˜ E X, G vol (cid:1) G ,vol , is neither a strong nor a weak H -valued discrete planarMarkovian holonomy field since it does not satisfy the weak independence property.Indeed, let us consider two loops l , l ′ and a path p as drawn in Figure 2 andlet us suppose that they are drawn on a finite planar graph G . Let us supposethat vol ( Int ( l )) = vol ( Int ( l ′ )) = 1. If (cid:0) ˜ E X, G vol (cid:1) G ,vol satisfied the weak independenceproperty, under ˜ E X, G vol , h ( l ) and h ( l ′ ) would be independent. Recall Equation (3.4),where f and f are, in this case, any functions on H since H is abelian. Under E X, G vol , the random couple ( h ( l ) , h ( pl ′ p − )) has the same law as ( U AU − , U BU − )where A , B and U are three independent random variables such that U is a Haarvariable on G and the two random variables A and B have the same law as X . Thetwo random variables U AU − and U BU − are not independent since the Equality(9.2) implies that: P (cid:2) ( U AU − , U BU − ) = ( c, c ) (cid:3) = P [ U AU − = c ] P [ U BU − = c ] . This proves that under ˜ E X, G vol , h ( l ) and h ( l ′ ) are not independent.Using Theorem 9.1, we can now prove Proposition 3.16. Proof of Proposition 3.16. As a consequence of Theorem 9.1, (cid:0) E G vol (cid:1) G ,vol is a discrete planar Yang-Mills field associated with a self-invariant by conjugationL´evy process ( Y t ) t ≥ . Besides, the law of a simple loop l under E G ( l ) vol is equalto R G m gvol ( Int ( l )) dg where m vol ( Int ( l )) is the law of Y vol ( Int ( l )) . Hence, under thehypothesis of Proposition 3.16, for any positive real t , Y t is almost surely in H : .2. CONSEQUENCES OF THE CHARACTERIZATION THEOREM 93 we can find a modification of ( Y t ) t ≥ which is H -valued. Using Theorem 7.11, wecan define a H -valued planar Yang-Mills field, associated with the H -valued L´evyprocess ( Y t ) t ≥ , whose restriction to planar graphs is a H -valued stochasticallycontinuous in law discrete planar Markovian holonomy field which satisfies therequired conditions. (cid:3) HAPTER 10 Classification of Stochastically Continuous StrongPlanar Markovian Holonomy Fields Let (cid:0) E vol (cid:1) vol be a stochastically continuous strong planar Markovian holo-nomy field: it is a planar Yang-Mills field to which is associated a L´evy process Y = ( Y t ) t ≥ . In Definition 7.13, the notions of pure non-degenerate/pure degener-ate/mixed degenerate planar Yang-Mills field were defined according to the degreeof symmetry and the support of Y . In this section, we will see equivalent conditionsfor (cid:0) E vol (cid:1) vol to be in each of these categories. The theorems explained below arestraightforward applications of Theorem 9.1 and Section 8.4.3. Indeed, by defini-tion, (cid:0) E vol (cid:1) vol is a pure non-degenerate (resp. pure) planar Yang-Mills field if andonly if Y is pure non-degenerate (resp. pure). Applying Proposition 8.24 (resp.Proposition 8.26, resp. Theorem 8.27) to the process Z t defined in Definition 9.7allows us to prove Theorem 10.1 (resp. Theorem 10.2, resp. Theorem 10.3).The first theorem gives an equivalent condition, when G is a finite group, for (cid:0) E vol (cid:1) vol to be a pure non-degenerate planar Yang-Mills field. Theorem . Let G be a finite group, let (cid:0) E vol (cid:1) vol be a stochastically contin-uous strong planar Markovian holonomy field. It is a pure non-degenerate planarYang-Mills field if and only if for any simple loop l , for any measure of area vol ,the support of h ( l ) under E vol is G . Let G be a compact Lie group and (cid:0) E vol (cid:1) vol be a G -valued stochastically con-tinuous strong planar Markovian holonomy field. The second theorem gives anequivalent condition for (cid:0) E vol (cid:1) vol to be a pure non-degenerate planar Yang-Millsfield. Theorem . Let us suppose that for any loop l and any measure of area vol , under (cid:0) E vol (cid:1) vol , e is in the support of h ( l ) . The planar Markovian holonomyfield (cid:0) E vol (cid:1) vol is a pure non-degenerate planar Yang-Mills field if and only if forany sequence of simple loops (cid:0) l n (cid:1) n ∈ N in R satisfying vol (cid:0) Int ( l n ) (cid:1) −→ n →∞ ∞ , one has: E vol ◦ h ( l n ) − −→ n →∞ λ G , where λ G is the Haar measure on G . One could remove the condition on the support of h ( l ) if one could understandthe support of any L´evy process which is invariant by conjugation. The thirdtheorem gives an equivalent condition for (cid:0) E vol (cid:1) vol to be a pure planar Yang-Millsfield. Theorem . The planar Markovian holonomy field (cid:0) E vol (cid:1) vol is a pure pla-nar Yang-Mills field if and only there exists a L´evy process ( X t ) t ≥ such that for 956 10. CLASSIFICATION OF PLANAR MARKOVIAN HOLONOMY FIELDS any simple loop l , for any measure of area vol , the law of h ( l ) under (cid:0) E vol (cid:1) vol is thelaw of X vol ( Int ( l )) . If this condition holds, then ( X t ) t ≥ is invariant by conjugationand it is the unique L´evy process associated with (cid:0) E vol (cid:1) vol . art 4 Markovian Holonomy Fields HAPTER 11 Markovian Holonomy Fields In this chapter, some definitions and results about Markovian holonomy fields,taken from [ L´ev10 ], are recalled. In the next chapter, the free boundary expec-tation on the plane associated with any Markovian holonomy field will be defined.This is a planar Markovian holonomy field which will allow us to apply the re-sults obtain previously and to characterize the spherical part of regular Markovianholonomy fields (Theorem 11.23). G -constraints Until the end of the paper, M will be an oriented smooth compact surface withboundary and, from now on, as we only consider such surfaces, we will call themsimply surfaces. Definition . To any connected component of the boundary of M one canassociate a non-oriented cycle (Definition 1.9). The union of these non-orientedcycles is denoted by B ( M ).A collection of marks C on M is a finite union of disjoint simple smooth non-oriented cycles in the interior of M . The couple ( M, C ) is called a marked surface and any element of C is called a mark .The orientation of M induces an orientation on each connected component ofthe boundary: we denote by B + ( M ) the subset of B ( M ) of positively orientedrepresentative of each non-oriented cycle. The non-oriented cycles included in C does not carry a canonical orientation.Let us recall that a non-oriented cycle is a set { c, c − } where c and c − areoriented cycle, thus by definition a mark is an oriented cycle. Besides if M has onlyone boundary, we will denote by ∂M the positively oriented cycle associated withthe unique boundary of M . Let ( M, C ) be a marked surface. Definition . A graph on ( M, C ) is a graph on M such that each orientedcycle in C is represented by a loop in L ( G ).The Proposition 1.3.10 in [ L´ev10 ] asserts that for any graph G on M , anycycle of B ( M ) is represented by a loop in L ( G ). Let G be a Lie group fixed oncefor all. Let Conj ( G ) be the set of conjugacy classes of G . Definition . A set of G -constraints on ( M, C ) is a mapping C from C ∪B ( M ) to Conj ( G ) such that C ( l − ) = C ( l ) − for any l ∈ C ∪ B ( M ). The family ofsets of G -constraints on ( M, C ) is denoted by Conj G ( M, C ). Notation . Let C be a set of G -constraints, let c be an oriented cycle in C ∪ B ( M ) and let x be an element of G . We will denote by C c → x the unique set of G -constraints such that: (1) for any oriented cycle c ′ ∈ C ∪ B ( M ) different of c and c − , C l → x ( c ′ ) = C ( c ′ ) , (2) C c → x ( c ) = [ x ] and C c → x ( c − ) = (cid:2) x − (cid:3) , where we recall that [ x ] is the conjugacy class of x . Besides, we will denote by c → [ x ] the set of G -constraints defined on { c, c − } which sends c on [ x ] and c − on [ x − ]. Definition . A measured marked surface with G -constraints is a quadru-ple ( M, vol, C , C ) where ( M, C ) is a marked surface, vol is a measure of area on M and C is a set of G -constraints on ( M, C ).The isomorphism notion on the set of measured marked surfaces with G -constraints is the following: ( M, vol, C , C ) and ( M ′ , vol ′ , C ′ , C ′ ) are isomorphic ifand only if there exists a diffeomorphism ψ : M → M ′ such that: • vol ◦ ψ − = vol ′ , • ψ sends C on C ′ , • ∀ l ∈ C ∪ B ( M ) , C ′ ( ψ ( l )) = C ( l ). An important notion for the definition of Markovian holonomy fields is theoperation of splitting. We will not define this notion rigorously in this paper butinstead we refer the reader to Section 1 . . L´ev10 ] for a rigorous definition.Let M be a surface, a splitting of M is the data of a surface M ′ and a gluing: M ′ → M , which is an application which glues two boundary components of M ′ .The set which consists of the image of the boundary glued and its inverse is thejoint of the gluing: we split according to this non-oriented cycle. Thus, we willconsider a splitting as the inverse of the gluing: a splitting is the action to splita surface according to a non-oriented cycle drawn on it. We will also say that wesplit a surface according to a mark l and by this, we mean that we split accordingto the non-oriented cycle { l, l − } . There is uniqueness (modulo isomorphism) ofthe splitting according to a mark on a surface M : the split surface of M accordingto l is denoted by Spl l ( M ).Let ( M, C , vol, C ) be a measured marked surface with G -constraints. Let l bea mark in C and f l : Spl l ( M ) → M be the gluing associated with the splitting Spl l ( M ). Thanks to the empty intersection of the marks on M , we can transportthe marks on Spl l ( M ). We will denote them Spl l ( C ). Since outside a negligiblesubset, a gluing is a diffeomorphism, it is possible to transport the measure of areaon Spl l ( M ) by setting Spl l ( vol ) = vol ◦ f l . In order to transport the G -constraintson Spl l ( M ), we set Spl l ( C )( l ′ ) = C ( f l ( l ′ )) for any l ′ ∈ Spl l ( C ) ∪ B ( Spl l ( M )). The definition of a Markovian holonomy field was first stated in Definition 3 . . L´ev10 ]. In this paper, we only consider oriented Markovian holonomy fields:for sake of simplicity, we will call them Markovian holonomy fields. In the followingdefinition, we add the condition that the measures are non-degenerate, which meansthat their weight is strictly positive. Besides, we change Axiom A : in order toundestand this change in the definition, one can read Remark 3.12. Definition . A G -valued Markovian holonomy field, HF , is the data, foreach measured marked surface with G -constraints ( M, vol, C , C ) of a non-degeneratefinite measure HF ( M,vol, C ,C ) on ( M ult ( P ( M ) , G ) , I ) such that: A : For any ( M, vol, C , C ), HF ( M,vol, C ,C ) ( ∃ l ∈ C ∪ B ( M ) , h ( l ) / ∈ C ( l )) = 0. A : For any ( M, vol, C ) and any event Γ in I , the function which sends C on HF ( M,vol, C ,C ) (Γ) is a measurable function on Conj G ( M, C ). A : For any ( M, vol, C , C ) and any l ∈ C , HF ( M,vol, C\{ l,l − } ,C |B ( M ) ∪C\{ l,l − } ) = Z G HF ( M,vol, C ,C l → [ x ] ) dx, where C l → [ x ] is defined in Notation 11.4. A : Let ψ : ( M, vol, C , C ) → ( M ′ , vol ′ , C ′ , C ′ ) be a bi-Lipschitz homeomor-phism which preserves the orientation such that vol ◦ ψ − = vol ′ , ψ ( C ) = C ′ and C = C ′ ◦ ψ . The mapping from M ult ( P ( M ′ ) , G ) to M ult ( P ( M ) , G )induced by ψ , also denoted ψ , satisfies: HF ( M ′ ,vol ′ , C ′ ,C ′ ) ◦ ψ − = HF ( M,vol, C ,C ) . Moreover, let G (resp. G ′ ) be a graph on ( M, C ) (resp. on ( M, C ′ )),let φ be a homeomorphism from ( M, vol, C , C ) to ( M ′ , vol ′ , C ′ , C ′ ) whichsends G on G ′ , which preserves the orientation and such that vol ◦ φ − = vol ′ , φ ( C ) = C ′ and C = C ′ ◦ φ . The mapping from M ult ( P ( G ′ ) , G ) to M ult ( P ( G ) , G ) induced by φ , also denoted φ , satisfies: (cid:0) HF ( M ′ ,vol ′ , C ′ ,C ′ ) (cid:1) |M ult ( P ( G ′ ) ,G ) ◦ φ − = (cid:0) HF ( M,vol, C ,C ) (cid:1) |M ult ( P ( G ) ,G ) . A : For any ( M , vol , C , C ) and ( M , vol , C , C ), one has the identity: HF ( M ⊔ M ,vol ⊔ vol , C ⊔C ,C ⊔ C ) = HF ( M ,vol , C ,C ) ⊗ HF ( M ,vol , C ,C ) . A : For any ( M, vol, C , C ), any l ∈ C and any gluing along l , ψ : Spl l ( M ) → M , one has: HF ( Spl l ( M ) ,Spl l ( vol ) ,Spl l ( C ) ,Spl l ( C )) = HF ( M,vol, C ,C ) ◦ ψ − . A : For any ( M, vol, ∅ , C ) and for any l in B ( M ), Z G HF ( M,vol, ∅ ,C l → x ) ( ) dx = 1 . The Markovian holonomy fields are easier to understand them when they areexposed in a less formal way. A Markovian holonomy field is a family of mea-sures . For each surface M with marks, set of G -constraints and measure of area,we are given a gauge-invariant random holonomy field on M which satisfies theset of G -constraints ( A ). Moreover, the family of measures given by a Markovianholonomy field is invariant under a class of area-preserving homeomorphisms , A ,and satisfies a kind of Markov property , A and A . The measures associated with( M, vol, C , C ), seen as a function of the G -constraints, provide a regular disinte-gration of HF ( M,vol, ∅ ,C |B ( M ) ) (Axioms A , A and A ). The last assumption is a normalization axiom.As for the planar Markovian holonomy fields, if not specified, all the Markovianholonomy fields will be G -valued, thus we will omit to specify it. In the definitionof a Markovian holonomy field, we didn’t specify any regularity condition on thefield. In what follows, we will focus only on regular Markovian holonomy field inthe following sense. 02 11. MARKOVIAN HOLONOMY FIELDS Definition . Let HF be a Markovian holonomy field.(1) HF is stochastically continuous if, for any ( M, vol, C , C ), HF ( M,vol, C ,C ) isstochastically continuous (Definition 1.20).(2) HF is Fellerian if, for any ( M, vol, C ), the function( t, C ) HF ( M,vol, C ,C ) ( ) , defined on R ∗ + × Conj G ( M, C ) is continuous.(3) HF is regular if it is both stochastically continuous and Fellerian. Given an even positive integer g and p a positive integer, let Σ + p,g be the con-nected sum of g tori with p holes. For g = 0 we define Σ + p, to be the sphere with p holes. The classification of surfaces asserts that any connected oriented compactsurface is diffeomorphic to one and exactly one of (cid:8) Σ + p,g , p ∈ N , g ∈ N (cid:9) . Besides,as a consequence of a theorem of Moser and as explained in Proposition 4.1.1 of[ L´ev10 ], if M and M ′ are oriented, if ( M, vol, ∅ , C ) and ( M ′ , vol ′ , ∅ , C ′ ) are twomeasured marked surfaces with G -constraints, then they are isomorphic if and onlyif: (1) M and M ′ are diffeomorphic,(2) vol ( M ) = vol ′ ( M ′ ) , (3) there exists a bijection ψ = B + ( M ) → B + ( M ′ ) such that C = C ′ ◦ ψ on B + ( M ) . Let HF be a Markovian holonomy field, we define the partition functions of HF . Definition . Let g be an even positive integer, p be a positive integerand t be a positive real. Let vol be a measure of area on Σ + p,g of total mass t . Let { b , b , ..., b p } be an enumeration of B + (Σ + p,g ). We define the mapping: Z + p,g,t ( x , ..., x p ) : G p −→ R ∗ + ( x , ..., x p ) Z + p,g,t ( x , ..., x p ) = HF ( Σ + p,g ,vol, ∅ , ( b i [ x i ]) pi =1 )( ) , It is called the partition function of the surface of genus g with p holes. Usingthe diffeomorphism invariance given by Axiom A and using Moser’s theorem, itdepends neither on the choice of vol nor on the choice of the enumeration: Z + p,g,t isa symmetric function. Remark . If p = 0, we define Z +0 ,g,t as the positive number which is equalto the mass of HF ( Σ +0 ,g ,vol, ∅ , ∅ ).The discussion on the notion of isomorphism between measured marked surfaceswith G -constraints implies that if ( M, vol, ∅ , C ) is a measured marked surface with G -constraints then there exist p and g such that M is diffeomorphic to Σ + p,g : HF ( M,vol, ∅ ,C ) ( ) = Z + p,g,vol ( M ) ( x , ..., x p ) , where x , ..., x p are representatives of the p constraints put on B + ( M ).The Fellerian condition satisfied by regular Markovian holonomy fields impliesthat their partition functions are continuous in ( t, x , ..., x p ). Besides we can re-formulate the axiom of normalization A (Definition 11.6) in terms of partition functions. If HF is a Markovian holonomy field, for any t > Z G Z +1 , ,t ( g ) dg = 1 , that is to say: Z +1 , ,t dg is a probability measure on G . In one of the main theo-rems proved in Chapter 4 of [ L´ev10 ], L´evy characterized the family of probabilitymeasures (cid:0) Z +1 , ,t dg (cid:1) t> . Theorem . Let HF be a regular Markovian holonomy field. The probabil-ity measures (cid:0) Z +1 , ,t dg (cid:1) t> on G are the one dimensional distributions of a uniqueconjugation-invariant L´evy process ( Y t ) t ≥ . Moreover, this L´evy process character-izes completely the partition functions of HF . We say that Y = ( Y t ) t ≥ (resp. HF ) is the L´evy process (resp. a regular Mar-kovian holonomy field) associated with HF (resp. to Y ). Given this theorem, itis natural to wonder if every L´evy process which is conjugation-invariant is asso-ciated with a regular Markovian holonomy field. Of course, some other conditionsmust hold such as the existence of a conjugation-invariant square-integrable den-sity. Indeed, as the constraints on the boundary are given by specifying a conjugacyclass, Z +1 , ,t ( x ) is a function of [ x ]. Besides, by definition of regularity, it must becontinuous in x thus square-integrable. To finish, let us remark that Z +1 , ,t ( x ) isstrictly positive since we supposed that the measures (cid:0) H ( M,vol, C ,C ) (cid:1) ( M,vol, C ,C ) arenon-degenerate. Hence the natural following definition: Definition . Let ( Y t ) t ≥ be a L´evy process on G . It is admissible if: • it is invariant by conjugation by G , • the distribution of Y t admits a strictly positive square-integrable density Q t with respect to the Haar measure on G for any t > Proposition . Let HF be a regular Markovian holonomy field, the L´evyprocess ( Y t ) t ≥ associated with HF is an admissible L´evy process. In fact, we get all the admissible L´evy processes by studying the L´evy processeswhich are associated with regular Markovian holonomy fields: this is given byTheorem 4 . . L´ev10 ]. Theorem . Every admissible L´evy process Y is associated with a regularMarkovian holonomy field. The proof of this assertion consists in constructing, just as we did for planarMarkovian holonomy fields, for every admissible L´evy, a special Markovian holo-nomy field YM which will be called a Yang-Mills field. For this, L´evy used the edgeparadigm for random holonomy fields. A Yang-Mills field is a kind of deformationof a uniform measure. Let ( M, vol, C , C ) be a measured marked surface with G -constraints, endowedwith a graph G = ( V , E , F ). The uniform measure on M ult ( P ( G ) , G ) is almost aproduct of Haar measures as for any orientation E + of G , M ult ( P ( G ) , G ) ≃ G E + . 04 11. MARKOVIAN HOLONOMY FIELDS But one has to be careful: since ( M, C , C ) is an oriented marked surface with G -constraints, the elements in M ult ( P ( G ) , G ) that we have to consider have to obeythe constraints. Notation . For any conjugacy class O ⊂ G and any integer n ≥ 1, wedenote by δ O ( n ) the natural extension to G n of the unique G n -invariant probabilitymeasure on O ( n ) = { ( x , ..., x n ) ∈ G n : x ...x n ∈ O} under the G n action ( g , ..., g n ) • ( x , ..., x n ) = ( g x g − , ..., g n x n g − ).Let l , ..., l q be q disjoint simple loops in L ( G ) such that C ∪ B ( M ) is equalto { l , l − , ..., l q , l − q } . For any i ∈ { , ..., q } , we can decompose l i = e i, ...e i,n i with e i,j ∈ E for any i and j . Let E + be an orientation of G , such that for any i ∈ { , ..., q } and j ∈ { , ..., n i } , e i,j ∈ E + . We label e , ..., e m the other edgesof E + . Recall that any measure constructed on G E + defines canonically a uniquemeasure on M ult ( P ( G ) , G ). Definition . The uniform measure U G M, C ,C is the measure provided bythe following measure on G E + : dg ⊗ ... ⊗ dg m ⊗ δ C ( l )( n ) ( dg ,n ...dg , ) ⊗ ... ⊗ δ C ( l q )( n q ) ( dg q,n q ...dg q, ) . This probability measure on M ult ( P ( G ) , G ) does not depend on any of the choiceswe made. Notation . We define also a similar measure without constraints for anysurface M (resp. R ) endowed with a graph (resp. a planar graph) G . Let E + bean orientation of G . The measure on M ult ( P ( G ) , G ) seen on G E + as N e ∈ E + dg e is denoted by U G .Yang-Mills fields can now be defined. Definition . Let ( Y t ) t ≥ be an admissible L´evy process on G . For anypositive real t , let Q t be the density of Y t . A regular Markovian holonomy field YM is called a Yang-Mills field (or sometimes a Yang-Mills measure) associated with( Y t ) t ≥ if for any measured marked surface with G -constraints ( M, vol, C , C ) andany graph G on ( M, C ), (cid:0) YM ( M,vol, C ,C ) (cid:1) |M ult ( P ( G ) ,G ) = Y F ∈ F Q vol ( F ) ( h ( ∂F )) U G M, C ,C ( dh ) , where ∂F is the oriented facial cycle associated with F , defined in Definition 1 . . L´ev10 ] and the notation Q vol ( F ) ( h ( ∂F )) means that we consider Q vol ( F ) ( h ( c ))where c represents ∂F : this does not depend on the choice of c since Q vol ( F ) isinvariant by conjugation.A Yang-Mills field associated with ( Y t ) t ≥ is a regular Markovian holonomyfield associated with ( Y t ) t ≥ : Theorem 11.13 is a consequence of the followingproposition. Proposition . For any G -valued admissible L´evy process ( Y t ) t ≥ thereexists a unique Yang-Mills field YM associated with ( Y t ) t ≥ . For this proposition one has to introduce, as we did for planar Markovian holo-nomy fields, a discrete analog of Markovian holonomy fields: the discrete Markovianholonomy fields. The definition of discrete Markovian holonomy fields can be foundin Section 3 . L´ev10 ]. Then one can show that the family of measures: (cid:16)(cid:0) YM ( M,vol, C ,C ) (cid:1) |M ult ( P ( G ) ,G ) (cid:17) ( M,vol, C ,C, G ) is a Fellerian continuously area-dependent (Proposition 4 . . 11 in [ L´ev10 ]) and lo-cally stochastically -H¨older continuous (Proposition 4 . . 15 in [ L´ev10 ]) discreteMarkovian holonomy field (Proposition 4.3.10 in [ L´ev10 ]) associated with Y . Thenit is shown, in Theorem 3.2.9 of [ L´ev10 ], that under these regularity conditions,every discrete Markovian holonomy field can be extended to a regular Markovianholonomy field. It has to be noticed that we changed the definition of Markovianholonomy fields (Axiom A ): this allows us to correct the arguments used in theproof of Axiom A in Theorem 3.2.9 of [ L´ev10 ] by using the one explained beforeTheorem 3.11. This allows to conclude for the proof of Proposition 11.18.The definition of discrete Markovian holonomy field follows closely the defini-tion of a Markovian holonomy field except for the invariance by homeomorphismswhich becomes almost a combinatorial condition. It is the same difference betweenthe Axioms P and DP of Definitions 3.1 and 3.4 in Section 3.1. Remark . The difference between the assumption A in Definition 3.1.2in [ L´ev10 ] and D in Definition 3.2.1 in [ L´ev10 ] makes the proof of Lemma 3.2.2.in the same book incomplete. Thus, it is not clear that any Markovian holonomyfield defines by restriction a discrete Markovian holonomy field.This remark leads us to the following definition. Definition . Let HF be a Markovian holonomy field. It is constructible if the family of measures (cid:16)(cid:0) HF ( M,vol, C ,C ) (cid:1) |M ult ( P ( G ) ,G ) (cid:17) ( M,vol, C ,C, G ) is a discreteMarkovian holonomy field.It is still an open question to know if any Markovian holonomy field is con-structible. We can resume the results of Proposition 11.12 and Theorem 11.13 by thefollowing diagram.Regular Markovian holonomy fields Partition function % % Admissible L´evy processes Yang-Mills fields e e Besides, it was shown that the left arrow goes into the constructible regularMarkovian holonomy fields and the composition of the two arrows is equal to theidentity on the set of admissible L´evy processes. It is natural to wonder if the twoarrows are each other inverse: this leads us to the following conjecture. 06 11. MARKOVIAN HOLONOMY FIELDS Conjecture . Every regular Markovian holonomy field is a Yang-Millsfield.From this conjecture, it would be true that every regular Markovian holonomyfield is constructible. In order to state the main result concerning this conjecture,we need the notion of planar mark. Let M be an oriented smooth compact surfacewith boundary. Definition . A planar mark is a mark l on M such that l cuts M in twoparts, one of which is of genus 0. Theorem . Let (cid:0) HF ( M,vol, C ,C ) (cid:1) ( M,vol, C ,C ) be a regular Markovian holo-nomy field and ( Y t ) t ∈ R + its associated G -valued L´evy process. Let us consider (cid:0) YM ( M,vol, C ,C ) (cid:1) M,vol, C ,C the Yang-Mills field associated with ( Y t ) t ∈ R + .Let ( M, vol, ∅ , C ) be a measured marked surface with G -constraints, let l be aplanar mark on M , let M be a part of M of genius defined by l and let m be apoint in M . The following equality holds: (cid:0) HF ( M,vol, ∅ ,C ) (cid:1) |M ult ( L m ( M ) ,G ) = (cid:0) YM ( M,vol, ∅ ,C ) (cid:1) |M ult ( L m ( M ) ,G ) . Let C be a collection of marks on M which do not intersect the mark l . Let us choosean orientation of C denoted by C + . Let C be a set of G -constraints on B ( M ) . Weendow the set of G -constraints on C ∪B ( M ) with the measure dλ C |B ( M ) coming from: O c ∈C + dg c ⊗ O b ∈B ( M ) + δ C ( b ) . By disintegration, for any set of constraints on B ( M ) , dλ C |B ( M ) almost surely: (cid:0) HF ( M,vol, C ,C ) (cid:1) |M ult ( L m ( M ) ,G ) = (cid:0) YM ( M,vol, C ,C ) (cid:1) |M ult ( L m ( M ) ,G ) . In order to prove Conjecture 11.21, one would have to generalize Theorem11.23 in order to include all the remaining loops, including the generators of thefundamental group of the surface.HAPTER 12 The Free Boundary Condition on The Plane Let HF be a regular Markovian holonomy field which will be fixed until the endof the chapter. The measure HF ( M,vol, C ,C ) is not in general a probability measure.One way to deal with probability measure would be to normalize it by their mass.Yet, a better way to get a probability measure is to consider the free boundarycondition measure. Let M be a surface homomorphic to a disk Σ +0 , endowed with a measure ofarea vol . Definition . The free boundary condition expectation on M associatedwith HF is the probability measure on (cid:0) M ult (cid:0) P ( M ) , G (cid:1) , B (cid:1) such that for any pos-itive integer n , any measurable function f : G n → R + and any finite family c , ... c n of elements of P ( M ): E HF M,vol (cid:16) f (cid:0) h ( c ) , ..., h ( c n ) (cid:1)(cid:17) = Z G c HF ( M,vol, ∅ ,∂M → [ x ]) (cid:16) f (cid:0) h ( c ) , ..., h ( c n ) (cid:1)(cid:17) dx, where c HF M,vol, ∅ ,∂M → [ x ] is the extension of HF M,vol, ∅ ,∂M → [ x ] to the Borel σ -fieldgiven by Proposition 1.30. Remark . In this definition we have extended the σ -field to the Borel σ -field, in a way such that the new measure becomes invariant by the gauge group.In order for the definition of E HF M,vol to be consistent with the way we named it, onehas to verify that it is indeed a probability measure. Since the constant function is gauge-invariant, ˆ J c ,...,cn = , thus E HF M,vol ( ) = R G HF ( M,vol, ∅ ,∂M → [ x ]) ( ) dx = 1 , the last equality coming from the normalization Axiom A in Definition 11.6.The free boundary condition expectation on M of a Yang-Mills field is computedin the following lemma. Lemma . Let YM be the Yang-Mills field associated with a G -valued ad-missible L´evy process ( Y t ) t ∈ R + . For any positive real t , let Q t be the density of Y t .Let G be a graph on M : (cid:0) E YM M,vol (cid:1) |M ult ( P ( G ) ,G ) = Y F ∈ F Q vol ( F ) ( h ( ∂F )) U G ( dh ) , where U G was defined in Notation 11.16 and where we used the notation Q vol ( F ) ( h ( ∂F )) already used in Proposition 7.6. Proof. This follows from the fact that R G U G M, ∅ ,∂M → x dx = U G which is aconsequence of: R G (cid:20) R G n f δ [ y ]( n ) ( dx , ..., dx n ) (cid:21) dy = R G n f dx ...dx n , given by theEquality (26) of Lemma 2 . . L´ev10 ]. (cid:3) Proposition . Let ( M, vol ) and ( M ′ , vol ′ ) be two measured compact two-dimensional sub-manifolds of R which are homeomorphic to the unit disk. Letus suppose that M is included in the interior of M ′ , denoted by Int ( M ′ ) . Let usassume that vol ′| M = vol . The free boundary condition expectations on M and M ′ are related by: E HF M,vol = E HF M ′ ,vol ′ ◦ ρ − M,M ′ , where we remind the reader that ρ M,M ′ was defined in Notation 1.21. Thus, for any measure of area vol on R , the family: (cid:26)(cid:16) M ult ( P ( M ) , G ) , B , E HF M,vol | M (cid:17) M ⊂ R , ( ρ M,M ′ ) M ⊂ Int ( M ′ ) (cid:27) , is a projective family of probability spaces. Proof. Since E HF M,vol and E HF M ′ ,vol ′ ◦ ρ − M,M ′ are gauge-invariant, it is enough,by Proposition 1.37, to show that, for any positive integer n , for any continuousconjugation-invariant function f on G n and any n -tuple of loops l , ..., l n in M basedat a fixed point m of M : E HF M ′ ,vol ′ (cid:2) f (cid:0) h ( l ) , ..., h ( l n ) (cid:1)(cid:3) = E HF M,vol (cid:2) f (cid:0) h ( l ) , ..., h ( l n ) (cid:1)(cid:3) . The l.h.s. is equal to: Z G Z M ( P ( M ′ ) ,G ) f (cid:0) h ( l ) , ..., h ( l n ) (cid:1) HF ( M ′ ,vol ′ , ∅ ,∂M ′ → [ x ]) ( dh ) dx = Z G Z G Z f (cid:0) h ( l ) , ..., h ( l n ) (cid:1) HF ( M ′ ,vol ′ ,∂M, { ∂M ′ → [ x ] ,∂M → [ y ] } ) ( dh ) dydx = Z G Z G Z M ( P (( M ′ \ Int ( M )) ⊔ M ) ,G ) f (cid:0) h ( l ) , ..., h ( l n ) (cid:1) HF (cid:16) ( M ′ \ Int ( M )) ⊔ M,vol ′| ( M ′\ Int ( M )) ⊔ vol, ∅ , { ∂M ′ → [ x ] ,∂M → [ y ] } (cid:17) ( dh ) dydx = Z G Z G Z M ( P ( M ′ \ Int ( M )) ,G ) Z M ( P ( M ) ,G ) f (cid:0) h ( l ) , ..., h ( l n ) (cid:1) HF ( M,vol, ∅ , { ∂M → [ y ] } ) ( dh ) HF (cid:16) M ′ \ Int ( M ) ,vol ′| M ′\ Int ( M ) , ∅ , { ∂M ′ → [ x ] ,∂M → [ y ] } (cid:17) ( dh ′ ) dydx = Z G Z G Z M ( P ( M ) ,G ) f (cid:0) h ( l ) , ..., h ( l n ) (cid:1) HF ( M,vol, ∅ , { ∂M → [ y ] } ) ( dh ) HF (cid:16) M ′ \ Int ( M ) ,vol ′| M ′\ Int ( M ) , ∅ , { ∂M ′ → [ x ] ,∂M → [ y ] } (cid:17) ( ) dydx = Z G Z M ( P ( M ) ,G ) f (cid:0) h ( l ) , ..., h ( l n ) (cid:1) HF ( M,vol, ∅ , { ∂M → [ y ] } ) ( dh ) (cid:16) Z G HF (cid:16) M ′ \ Int ( M ) ,vol ′| M ′\ Int ( M ) , ∅ , { ∂M ′ → [ x ] ,∂M → [ y ] } (cid:17) ( ) dx (cid:17) dy = Z G Z M ( P ( M ) ,G ) f (cid:0) h ( l ) , ..., h ( l n ) (cid:1) HF ( M,vol, ∅ , { ∂M → [ y ] } ) ( dh ) dy = E HF M,vol h f (cid:0) h ( l ) , ..., h ( l n ) (cid:1)i , where we applied successively the definition of E HF M ′ ,vol ′ , the Axioms A , A and A . Then after a change of notation and a Fubini exchange of integrals, the nor-malization Axiom A with the definition of E HF M,vol lead us to the result. (cid:3) The free boundary expectation on the plane is the projective limit of this familyof measured spaces. Let vol be a measure of area on R . Definition . The free boundary condition expectation on R associatedwith HF , denoted by E HF vol , and defined on (cid:0) M ult ( P ( R ) , G ) , B (cid:1) is the projectivelimit of: (cid:26)(cid:16) M ult ( P ( M ) , G ) , B , E HF M,vol | M (cid:17) M ⊂ R , ( ρ M,M ′ ) M ⊂ Int ( M ′ ) (cid:27) . This random holonomy field is gauge-invariant.Lemma 12.3 gives for any embedded planar graph G an explicit formula forthe restriction on M ult ( P ( G ) , G ) of the free boundary condition expectation onthe plane associated with a Yang-Mills field. Proposition 2.7 asserts that any finiteplanar graph G ′ can be seen as a subgraph of an embedded planar graph. It isthus possible to give an explicit formula for the restriction on M ult ( P ( G ′ ) , G ) ofthe free boundary condition expectation on the plane associated with a Yang-Millsfield. Proposition . Suppose that R is endowed with a measure of area vol .Let G = ( V , E , F ) be a finite planar graph. Let Y = ( Y t ) t ≥ be a G -valued admissibleL´evy process with associated semigroup of densities ( Q t ) t ≥ . Let YM be the Yang-Mills field associated with Y . The free boundary condition expectation on R of YM satisfies: (cid:0) E YM vol (cid:1) |M ult ( P ( G ) ,G ) ( dh ) = Y F ∈ F b Q vol ( F ) (cid:0) h ( ∂F ) (cid:1) U G ( dh ) , where ∂F is the anti-clockwise oriented facial cycle associated with F and where weused the same convention as before for Q vol ( F ) (cid:0) h ( ∂F ) (cid:1) . In order to simplify the proof, we will use the upcoming Theorem 12.8. Proof. We have already seen in the proof of Proposition 7.6 that Y F ∈ F b Q vol ( F ) (cid:0) h ( ∂F ) (cid:1) U G ( dh ) G ,vol is a stochastically continuous in law weak discrete planar Markovian holonomyfield. Using Theorem 12.8 and the constructibility result of Section 4, the family (cid:16)(cid:0) E YM vol (cid:1) |M ult ( P ( G ) ,G ) (cid:17) G ,vol is a stochastically continuous in law weak discrete planarMarkovian holonomy field. Recall the definition of L i,j in Definition 5.9. UsingTheorem 9.6, we only have to check that for any positive real α , (cid:0) h ( L n, ) (cid:1) n ∈ N hasthe same law under E YM αdx as under Q F ∈ F b Q α (cid:0) h ( ∂F ) (cid:1) U N ( dh ), where F b is the setof bounded faces of the N graph. The value of α will not be important, so wewill suppose that α = 1. In fact, we will prove that for any positive integer n , (cid:0) E YM dx (cid:1) |M ult ( P ( G n ) ,G ) ( dh ) = Q F ∈ F bn Q ( h ( ∂F )) U G n ( dh ), where G n = ( V n , E n , F n ) = N ∩ (cid:0) [0 , n ] × [0 , (cid:1) . 10 12. THE FREE BOUNDARY CONDITION ON THE PLANE Let ∂D (0 , n + 1) be the loop based at ( n + 1 , 0) turning anti-clockwise, rep-resenting the cycle bounding the disk of radius n + 1 centered at (0 , G ′ n = ( V ′ n , E ′ n , F ′ n ) be the graph defined by: • E ′ n = E n ∪ (cid:8) ∂D (0 , n + 1) , ∂D (0 , n + 1) − , e rn, , ( e rn, ) − (cid:9) , • V ′ n = V n ∪ (cid:8) ( n + 1 , (cid:9) .The finite planar graph G ′ n is an embedded graph and G n is a subgraph of G ′ n .Using Lemma 12.3: (cid:0) E YM dx (cid:1) |M ult ( P ( G ′ n ) ,G ) ( dh ) = Y F ∈ F ′ bn Q dx ( F ) (cid:0) h ( ∂F ) (cid:1) U G ′ n ( dh ) . Since (cid:0)Q F ∈ F b Q vol ( F ) (cid:0) h ( ∂F ) (cid:1) U G ( dh ) (cid:1) G ,vol is a weak discrete planar Markovian ho-lonomy field, the restriction of Q F ∈ F ′ bn Q dx ( F ) (cid:0) h ( ∂F ) (cid:1) U G ′ n ( dh ) to M ult ( P ( G n ) , G )is Q F ∈ F bn Q ( h ( ∂F )) U G n ( dh ): (cid:0) E YM dx (cid:1) |M ult ( P ( G n ) ,G ) ( dh ) = Q F ∈ F bn Q ( h ( ∂F )) U G n ( dh ) . (cid:3) This last proposition and Proposition 7.6 show that the free boundary conditionexpectation on R of a Yang-Mills field associated with an admissible L´evy process Y is the planar Yang-Mills field associated with Y . This implies the following result. Proposition . Let YM be the Yang-Mills field associated with an admis-sible L´evy process Y = ( Y t ) t ∈ R + . For any planar graph G = ( V , E , F ) , any measureof area vol , any family of facial loops ( c F ) F ∈ F b oriented anti-clockwise and anyrooted spanning tree T of G , under the free boundary condition on the plane E YM vol ,the random variables (cid:0) h ( l c F ,T ) (cid:1) F ∈ F b are independent and for any F ∈ F b , h ( l c F ,T ) has the same law as Y vol ( F ) . The free boundary condition expectation on R allows us to link the theory ofMarkovian holonomy fields with the one of planar Markovian holonomy fields. Con-sider HF a regular Markovian holonomy field and let (cid:0) E HF vol (cid:1) vol be the free boundarycondition expectation on the plane associated with HF . Theorem . The family (cid:0) E HF vol (cid:1) vol is a stochastically continuous strong pla-nar Markovian holonomy field. Using the theory of planar Markovian holonomy fields, it is enough to showthat for any vol , E HF vol is stochastically continuous and that its restriction to Aff ( R )is a stochastically continuous weak planar Markovian holonomy field. As we havealready checked the weight condition and as we have noticed the gauge-invarianceof the free boundary condition expectation in Definition 12.5, it remains to showthat it is stochastically continuous and that the Axioms wP , wP and wP inDefinition 3.2 hold. These are proved in the following Lemmas 12.9, 12.10, 12.11and 12.12. Lemma . For any measure of area vol , E HF vol is a stochastically continuousrandom holonomy field. Proof. Let vol be a measure of area, let p n be a sequence of paths whichconverges, as n goes to infinity, to a path p for the convergence with fixed endpoints.Let D be a disk centered at (0 , 0) such that for any integer n , p n ∈ D . We remindthe reader that c HF ( D ,vol | D , ∅ ,∂ D → [ x ] ) is the extension given by Proposition 1.30 of HF ( D ,vol | D , ∅ ,∂ D → [ x ] ) on the Borel σ -field. By definition, E HF vol (cid:2) d G ( h ( p n ) , h ( p ) (cid:3) = Z G c HF ( D ,vol | D , ∅ ,∂ D → [ x ] ) (cid:2) d G ( h ( p n ) , h ( p )) (cid:3) dx = Z G HF ( D ,vol | D , ∅ ,∂ D → [ x ] ) h d ( d G ) J { pn,p } ( h ( p n ) , h ( p )) i dx. Since p n and p have the same endpoints, J { p n ,p } is equal to G and its action on G is given by: ( k , k ) • ( g , g ) = ( k − g k , k − g k ). The invariance of d G , byright and left translations, implies that d ( d G ) J { pn,p } = d G . This leads to: E HF vol (cid:2) d G ( h ( p n ) , h ( p ) (cid:3) = Z G HF ( D ,vol | D , ∅ ,∂ D → [ x ] ) (cid:2) d G ( h ( p n ) , h ( p )) (cid:3) dx. Since HF is regular, it is stochastically continuous, thus we have: HF ( D ,vol | D , ∅ ,∂ D → [ x ] ) (cid:2) d G ( h ( p n ) , h ( p )) (cid:3) −→ n →∞ . Thus, with an argument of dominated convergence, E HF vol (cid:2) d G ( h ( p n ) , h ( p )) (cid:3) convergesto zero as n goes to infinity. (cid:3) Lemma . The family of random holonomy fields (cid:0) E HF vol (cid:1) vol satisfies thearea-preserving diffeomorphisms at infinity invariance property wP . Proof. Consider vol and vol ′ two measures of area on R . Let ψ be a diffeo-morphism at infinity which preserves the orientation and let R be a positive realsuch that:(1) vol ′ = vol ◦ ψ − ,(2) ψ : D (0 , R ) c → ψ (cid:0) D (0 , R ) c (cid:1) is a diffeomorphism.Using the gauge-invariance of E HF vol , it is enough to consider piecewise affine loopsbased at the same point. Let l , ..., l n be loops in Aff (cid:0) R (cid:1) based at the same pointsuch that for any i ∈ { , ..., n } , l ′ i = ψ ( l i ) is in Aff (cid:0) R (cid:1) . Let R ′ be a positivereal such that R ′ is greater than R and such that for any i ∈ { , ..., n } , l i is in M R ′ = D (0 , R ′ ). The set M ′ = ψ ( M R ′ ) is a connected compact two-dimensionalsub-manifold of R . Let us consider f : G n → R , a continuous function invariantby diagonal conjugation: E HF vol h f (cid:0) ( h ( l i ) ni =1 ) (cid:1)i is equal to: E HF M R ′ ,vol | MR ′ h f (cid:0) ( h ( l i ) ni =1 ) (cid:1)i = Z G Z M ult ( P ( M R ) ,G ) f (cid:16)(cid:0) h ( l i ) ni =1 (cid:1)(cid:17) HF (cid:16) M R ′ ,vol | MR ′ , ∅ ,∂M R → [ x ] (cid:17) ( dh ) dx = Z G Z M ult ( P ( M ′ ) ,G ) f (cid:16)(cid:0) h ( l ′ i ) ni =1 (cid:1)(cid:17) HF (cid:16) M ′ ,vol ′| M ′ , ∅ ,∂M ′ → [ x ] (cid:17) ( dh ) dx = E HF M ′ ,vol ′| M ′ h f (cid:0) ( h ( l ′ i ) ni =1 ) (cid:1)i = E HF vol ′ h f (cid:0) ( h ( l ′ i ) ni =1 ) (cid:1)i . The Axiom wP is satisfied by (cid:0) E HF vol (cid:1) vol . (cid:3) 12 12. THE FREE BOUNDARY CONDITION ON THE PLANE Lemma . The family of random holonomy fields (cid:0) E HF vol (cid:1) vol satisfies theweak independence property wP . Proof. Let vol be a measure of area on R . Let l and l ′ be two loops in Aff ( R ) such that Int ( l ) ∩ Int ( l ′ ) = ∅ . We can always consider ˜ l and ˜ l ′ two smoothsimple loops in R such that the closure of their interiors are also disjoint and suchthat l ⊂ Int (˜ l ) and l ′ ⊂ Int (˜ l ′ ). Using this remark, we can suppose that l and l ′ aresmooth. Using the gauge-invariance of E HF vol , as we did in order to show the Axiom wDP in the proof of Proposition 7.5, we can work with loops. Let us consider l , ..., l n some loops in Int ( l ) and l ′ , ..., l ′ m some loops in Int ( l ′ ). The aim is to provethat for any continuous functions f and g , from G n , respectively G m , to R , wehave: E HF vol h f (cid:0) ( h ( l i )) ni =1 (cid:1) g (cid:0) ( h ( l ′ i )) mi =1 (cid:1)i = E HF vol h f (cid:0) ( h ( l i )) ni =1 (cid:1)i E HF vol h g (cid:0) ( h ( l ′ i )) mi =1 (cid:1)i . We will use the notations and results stated in Remark 1.32. Let L be asmooth loop such that L surrounds l and l ′ . Depending on the context L weeither stand for Int ( L ) or for the oriented cycle represented by L . Besides, wewill suppose that the orientation of L was chosen such that L = ∂ Int ( L ). Thesame notations will hold for l and l ′ . Using the different axioms in Definition 11.6,we have: E HF dx h f (cid:0) ( h ( l i )) ni =1 (cid:1) g (cid:0) ( h ( l ′ i )) mi =1 (cid:1)i = E HF L ,dx h f (cid:0) ( h ( l i )) ni =1 (cid:1) g (( h ( l ′ i )) mi =1 ) i = Z G Z M ult ( P ( L ) ,G ) \ ( f ⊗ g ) J ( li ) ni =1 , ( l ′ i ) mi =1 (cid:0) ( h ( l i )) ni =1 , ( h ( l ′ i )) mi =1 (cid:1) HF ( L ,dx, ∅ , L → [ y ]) ( dh ) dy = Z G Z M ult ( P ( L ) ,G ) b f J ( li ) ni =1 (cid:0) ( h ( l i ) (cid:1) ni =1 (cid:1) b g J ( l ′ i ) mi =1 (cid:0) ( h ( l ′ i ) (cid:1) mi =1 (cid:1) HF ( L ,dx, ∅ , L → [ y ]) ( dh ) dy = Z G Z M ult ( P ( L ) ,G ) b f J ( li ) ni =1 (cid:0) ( h ( l i ) (cid:1) ni =1 (cid:1) b g J ( l ′ i ) mi =1 (cid:0) ( h ( l ′ i )) mi =1 (cid:1) HF ( L ,dx, ∅ , {L → [ y ] ,l ′ → [ z ] ,l → [ w ] } ) ( dh ) dydzdw = Z G Z M ult ( P ( l ) ,G ) b f J ( li ) ni =1 (cid:0) ( h ( l i )) ni =1 (cid:1) HF ( l,dx, ∅ ,l → [ w ]) ( dh ) Z M ult ( P ( l ′ ) ,G ) b g J ( l ′ i ) mi =1 (cid:0) ( h ( l ′ i )) ni =1 (cid:1) HF ( l ′ ,dx, ∅ ,l ′ → [ z ]) ( dh ) Z M ult ( P ( L \ ( l ∪ l ′ )) ,G ) HF ( L ,dx, ∅ , {L → [ y ] ,l ′ → [ z ] ,l → [ w ] } ) ( dh ) dydzdw. Since R G R M ult ( P ( L \ ( l ∪ l ′ )) ,G ) HF ( L ,dx, ∅ , {L → [ y ] ,l ′ → [ z ] ,l → [ w ] } ) ( dh ) dy is equal to 1, E HF dx h f (cid:0) ( h ( l i )) ni =1 (cid:1) g (cid:0) ( h ( l ′ i )) mi =1 (cid:1)i is equal to Z G Z M ult ( P ( l ) ,G ) ˆ f J ( li ) ni =1 (cid:0) ( h ( l i )) ni =1 (cid:1) HF ( l,dx, ∅ ,l → [ w ]) ( dh ) dw Z G Z M ult ( P ( l ′ ) ,G ) ˆ g J ( l ′ i ) mi =1 (cid:0) ( h ( l ′ i )) mi =1 (cid:1) HF ( l ′ ,dx, ∅ ,l ′ → [ z ]) ( dh ) dz, which is equal to E HF dx h f (cid:0) ( h ( l i )) ni =1 (cid:1)i E HF dx h g (cid:0) ( h ( l ′ i )) mi =1 (cid:1)i . (cid:3) Lemma . The family of random holonomy fields (cid:0) E HF vol (cid:1) vol satisfies thelocality property wP . Proof. Let l be a simple loop, let vol and vol ′ be two measures of area whoserestrictions to the closure of the interior of l are equal. The random holonomy fields E HF vol and E HF vol ′ being gauge invariant and stochastically continuous, by Proposition1.37, we only have to prove, for any loops l , ..., l n in Int ( l ) based at the same pointand for any continuous function f : G n → R invariant under the diagonal action of G , that: E HF vol h f (cid:0) h ( l ) , ..., h ( l n ) (cid:1)i = E HF vol ′ h f (cid:0) h ( l ) , ..., h ( l n ) (cid:1)i . Using Riemann’s uniformization theorem, we can find a smooth curve ˜ l in theinterior of l such that l , ..., l n are in the interior of ˜ l . Let M be the closure of theinterior of ˜ l : E HF vol h f (cid:0) h ( l ) , ..., h ( l n ) (cid:1)i = E HF M,vol | M h f (cid:0) h ( l ) , ..., h ( l n ) (cid:1)i = E HF M,vol ′| M h f (cid:0) h ( l ) , ..., h ( l n ) (cid:1)i = E HF vol ′ h f (cid:0) h ( l ) , ..., h ( l n ) (cid:1)i . This allows us to conclude. (cid:3) Remark . Using the same kind of calculations as the one explained inthis subsection and using Theorem 11.10, it is easy to see that for any simple loop l , the law of h ( l ) under E HF vol is the law of Y vol ( Int ( l )) where ( Y t ) t ∈ R + is the L´evyprocess associated with HF .HAPTER 13 Characterization of the Spherical Part of RegularMarkovian Holonomy Fields We have now all the tools in order to prove Theorem 11.23. Proof of Theorem 11.23. Let us remark that the second part about marksis a consequence of the first part by conditioning: we will prove the first assertion.Let (cid:0) HF ( M,vol, C ,C ) (cid:1) ( M,vol, C ,C ) be a regular Markovian holonomy field and ( Y t ) t ∈ R + its associated G -valued L´evy process. Let (cid:0) YM ( M,vol, C ,C ) (cid:1) M,vol, C ,C be the Yang-Mills field associated with ( Y t ) t ∈ R + .Let ( M, vol, ∅ , C ) be a measured marked surface with G -constraints, let l be aplanar mark on M , let M be a part of M of genius 0 defined by l and let m be apoint in M . We want to prove that: (cid:0) HF ( M,vol, ∅ ,C ) (cid:1) |M ult ( L m ( M ) ,G ) = (cid:0) YM ( M,vol, ∅ ,C ) (cid:1) |M ult ( L m ( M ) ,G ) . For any loops l , ..., l n in M based at m and any continuous function f invariantby diagonal conjugation, R f ( h ( l ) , ..., h ( l n )) HF ( M,vol, ∅ ,C ) ( dh ) is equal to: Z Z f ( h ( l ) , ..., h ( l n )) HF ( M ,vol | M , ∅ ,C | ∂M \{ l,l − } ∪{ l → [ x ] } ) ( dh ) HF M ,vol | M , ∅ ,C | ∂M \{ l,l − } ∪{ l → [ x ] } ( ) dx where M is the second part of M defined by l . Using Theorem 11.10, HF and YM have the same partition functions. Thus, since l is a planar mark, it is enough toshow that for any measure marked surface with G -constraints ( M, vol, ∅ , C ) suchthat M is homeomorphic to a sphere with a positive number p of holes, HF ( M,vol, ∅ ,C ) = YM ( M,vol, ∅ ,C ) . The proof can be made by induction on the number of holes: we will onlyprove the case where p = 1 since the arguments for the induction are similar.Let (cid:0) E HF vol (cid:1) vol be the free boundary condition expectation on the plane, definedin Definition 12.5, associated with HF . It is a stochastically continuous strongplanar Markovian holonomy field as shown in Theorem 12.8. Hence, by Theorem4.2, it induces a stochastically continuous in law weak discrete planar Markovianholonomy field (cid:16) E HF , G vol (cid:17) G ,vol . The Remark 12.13 ensures that the condition in orderto apply Theorem 10.3 is satisfied by (cid:16) E HF , G vol (cid:17) G ,vol . It is equal to the pure discreteplanar Yang-Mills field, denoted by (cid:16) E Y, G vol (cid:17) G ,vol , associated with the L´evy process( Y t ) t ∈ R + . By stochastic continuity, for any measure of area vol , E HF vol = E Yvol , where E Yvol is the pure continuous planar Yang-Mills field associated with ( Y t ) t ∈ R + . Let (cid:0) E YM vol (cid:1) vol be the associated free boundary condition expectation on theplane associated with YM . Using Proposition 12.7, for any measure of area vol , E YM vol = E Yvol . Recall the notation for the free boundary condition on a surface andlet us consider a disk-shaped suface M endowed with a measure of area vol . Thelast two equalities imply that E HF M,vol = E YM M,vol . Using Definition 12.1: E HF M,vol = Z G c HF ( M,vol, ∅ , { ∂M → [ x ] } ) dx, and a similar equation holds for YM . Let t be equal to vol ( M ) and let us define Z t ( x ) = HF ( M,vol, ∅ , { ∂M → [ x ] } ) ( ) which is also equal to YM ( M,vol, ∅ , { ∂M → [ x ] } ) ( ) andwhich is strictly positive. Then: E HF M,vol = Z G c HF ( M,vol, ∅ , { ∂M → [ x ] } ) Z t ( x ) Z t ( x ) dx. Besides, the law of h ( ∂M ) is Z t ( g ) dg : it implies that c HF ( M,vol, ∅ , { ∂M → [ x ] } ) Z t ( x ) is a disin-tegration of E HF M,vol with respect to h ( ∂M ). The same discussion holds for YM . Byalmost sure uniqueness of the disintegration we have: c HF ( M,vol, ∅ , { ∂M → [ x ] } ) Z t ( x ) = d YM ( M,vol, ∅ , { ∂M → [ x ] } ) Z t ( x ) , a.s. in x, thus: c HF ( M,vol, ∅ , { ∂M → [ x ] } ) = d YM ( M,vol, ∅ , { ∂M → [ x ] } ) , a.s. in x. (13.1)It remains to remove the a.s. part. Using Proposition 1.37 and Lemma 1.13, weneed to show that, for any continuous function f invariant by diagonal conjugationfrom G n to G and any piecewise affine loops, for any Riemannian metric, l , ..., l n in the interior of M , based at the same point: c HF ( M,vol, ∅ , { ∂M → [ x ] } ) ( f ( h ( l ) , ..., h ( l n )))= d YM ( M,vol, ∅ , { ∂M → [ x ] } ) ( f ( h ( l ) , ..., h ( l n ))) . Yet, given such n -tuple, we can always find a mark l such that l , ..., l n is in theinterior of l . Thus, it is enough to show that for any mark l , for any x ∈ G , oncewe restrain the measures on M ult ( P ( Int ( l )) , G ), we have the equality: c HF ( M,vol, ∅ , { ∂M → [ x ] } ) = d YM ( M,vol, ∅ , { ∂M → [ x ] } ) . Let l be a mark on M and let us denote by M ′ the closure of the interiorof l . Let us suppose that the orientation of l is such that l = ∂M ′ . Let us recallthat Z +2 , ,s was the notation for the partition function of the regular Markovianholonomy field HF associated to the planar annulus of total volume which is equalto s . Applying the Axioms A , A and A , we get that for any continuous function 3. SPHERICAL PART OF REGULAR MARKOVIAN HOLONOMY FIELDS 117 f invariant by diagonal conjugation from G n to G and any loops l , ..., l n in M ′ : c HF ( M,vol, ∅ , { ∂M → [ x ] } ) ( f ( h ( l ) , ..., h ( l n )))= Z f ( h ( l ) , ..., h ( l n )) HF ( M,vol, ∅ , { ∂M → [ x ] } ) ( dh )= Z G Z f ( h ( l ) , ..., h ( l n )) HF ( M,vol,l, { ∂M → [ x ] ,l → [ y ] } ) ( dh ) dy = Z G Z f ( h ( l ) , ..., h ( l n )) HF ( M ′ ,vol | M ′ , ∅ , { ∂M ′ → [ y ] } ) ⊗ HF ( M \ Int ( M ′ ) ,vol | M \ Int ( M ′ ) , ∅ , { ∂M → [ x ] ,∂M ′ → [ y ] } )( dh ) dy = Z G Z f ( h ( l ) , ..., h ( l n )) HF ( M ′ ,vol | M ′ , ∅ , { ∂M ′ → [ y ] } )( dh ) Z +2 , ,vol ( M \ Int ( M ′ )) ( x, y − ) dy. Thus, if we only consider the restriction on M ult ( P ( M ′ ) , G ): c HF ( M,vol, ∅ , { ∂M → [ x ] } ) = Z G c HF ( M ′ ,vol | M ′ , ∅ , { ∂M ′ → [ y ] } ) Z +2 , ,vol ( M \ Int ( M ′ )) ( x, y − ) dy. Recall that the partition functions of HF and YM are equal. Thus, again if we onlyconsider the restriction on M ult ( P ( M ′ ) , G ): d YM ( M,vol, ∅ , { ∂M → [ x ] } ) = Z G d YM ( M ′ ,vol | M ′ , ∅ , { ∂M ′ → [ y ] } ) Z +2 , ,vol ( M \ Int ( M ′ )) ( x, y − ) dy. Once we restrain the measures on M ult ( P ( M ′ ) , G )), using Equation (13.1), forany x ∈ G : c HF ( M,vol, ∅ , { ∂M → [ x ] } ) = Z G c HF ( M ′ ,vol | M ′ , ∅ , { ∂M ′ → [ y ] } ) Z +2 , ,vol ( M \ Int ( M ′ )) ( x, y − ) dy = Z G d YM ( M ′ ,vol | M ′ , ∅ , { ∂M ′ → [ y ] } ) Z +2 , ,vol ( M \ Int ( M ′ )) ( x, y − ) dy = d YM ( M,vol, ∅ , { ∂M → [ x ] } ) . This proves that the Equality (13.1) now holds for any x in G . (cid:3) . S P H E R I C A L P A R T O F R E G U L A R M A R K O V I ANH O L O N O M Y F I E L D S Planar Objects General Objects Strong Restriction ( ( Weak Thm. 4.2 (cid:9) (cid:9) Free boundary expectation Thm. 12.8 j j Regular Markovian H.F. Conj. 11.21 (cid:3) (cid:3) Partition function, Thm. 11.10 (cid:4) (cid:4) Def. 12.5 k k Weak discrete Thm. 9.1 (cid:9) (cid:9) Strong discrete Thm. 3.11 C C Planar Y.M. Thm. 7.11 h h Y.M. measures Def. 11.17 C C L´evy processes Self-invariant by conjugation Thm. 7.11 J J Admissible is j j Prop. 11.18 ; ; ibliography [AHKH86a] S. Albeverio, R. Høegh-Krohn, and H. Holden, Random fields with values in Liegroups and Higgs fields , Lecture notes in Phys. (1986), 1–13.[AHKH86b] , Stochastic Lie group-valued measures and their relations to stochastic curveintegrals, gauge fields and Markov cosurfaces , Springer, 1986.[AHKH88a] , Stochastic Multiplicative Measures, Generalized Markov Semigroups, andGroup-Valued Stochastic Processes and Fields , J. Funct. Anal. (1988), 154–184.[AHKH88b] , Stochastic multiplicative measures, generalized Markov semigroups, andgroup-valued stochastic processes and fields , J. Funct. Anal. (1988), no. 1, 154–184.[Art25] E. Artin, Theorie der Z¨opfe , Abh. Math. Sem. Hamburgischen Univ. (1925), 47–72.[Art47] , Theory of braids , Ann. of Math. (1947), no. 1, 101–126.[AS12] M. Anshelevich and A.N. Sengupta, Quantum free Yang-Mills on the plane , J. Geom.Phys. (2012), 330–343.[ASS93] B. Aronov, R. Seidel, and D. Souvaine, On compatible triangulations of simple poly-gons , Comput. Geom. (1993), 27–35.[Bae94] J.C. Baez, Generalized measures in gauge theory , Lett. Math. Phys. (1994), no. 3,213–223.[CDG16] G. C´ebron, A. Dahlqvist, and F. Gabriel, The generalized master fields ,arxiv.org/abs/1601.00214 (2016), 1–33.[DM90] B. Dacorogna and J. Moser, On a partial differential equation involving the Jacobiandeterminant , Ann. Inst. Henry Poincar´e, section C (1990), no. 1, 1–26.[Dri89] B.K. Driver, Y M : continuum expectations, lattice convergence, and lassos , Comm.Math. Phys. (1989), no. 4, 575–616.[Dri91] , Two-dimensional Euclidean quantized Yang-Mills fields , Probability modelsin mathematical physics, World Sci. Publ., 1991, pp. 21–36.[Gro85] L. Gross, A Poincar´e lemma for connection forms , J. Funct. Anal. (1985), no. 1,1–46.[Gro88] , The Maxwell equations for Yang-Mills theory , CMS Conf. Proc., vol. 9,Amer. Math. Soc., 1988, pp. 193–203.[Kal05] O. Kallenberg, Probabilistic symmetries and invariance principles , Probab. Appl.,Springer, 2005.[KI40] Y. Kawada and K. Itˆo, On the probability distribution on a compact group , Proc. ofthe Phy.-Math. Soc. of Japan (1940), no. 12, 977–998.[L´ev00] T. L´evy, Construction et ´etude `a l’´echelle microscopique de la mesure de Yang-Millssur les surfaces compactes , C. R. Acad. Sci. Paris Ser. I Math. (2000), no. 11,1019–1024.[L´ev03] , Yang-Mills Measure on compact surfaces , no. 790, American MathematicalSociety, 2003.[L´ev10] , Two-dimensional Markovian holonomy fields , no. 329, Ast´erisque, 2010.[L´ev12] , The master field on the plane , arxiv.org/abs/1112.2452 (2012), 1–133.[MT01] B. Mohar and C. Thomassen, Graphs on surfaces , Johns Hopkins University Press,2001.[SBG79] J.P. Serre, M. Buhler, and C. Goldstein, Groupes finis : cours `a l’E.N.S.J.F,1978/1979 , Montrouge : Ecole normale sup´erieure de jeunes filles, 1979.[Sen92] A.N. Sengupta, The Yang-Mills measure for S , J. Funct. Anal. (1992), no. 2,231–273. [Sen97] , Gauge theory on compact surfaces , Mem. Amer. Math. Soc. (1997),no. 600, viii+85.[Str60] K. Stromberg, Probabilities on a compact group , Trans. Amer. Math. Soc. (1960),no. 2, 295–309.[Vog81] A. Vogt, The isoperimetric inequality for curves with self-intersections , Canad. Math.Bull. (1981), 161–167.[Wit41] E. Witten, On quantum gauge theories in two dimensions , Comm. Math. Phys. (141), no. 1991, 153–209.[Wit92] , Two-dimensional gauge theories revisited , J. Geom. Phys. (1992), no. 4,303–368.[YM54] C.N. Yang and R.L. Mills, Conservation of isotopic spin and isotopic gauge invari-ance , Physical Rev.96