QQuenched local convergence of Boltzmann planarmaps
Benedikt StuflerAbstract
Stephenson (2018) established annealed local convergence of Boltz-mann planar maps conditioned to be large. The present work uses results onrerooted multi-type branching trees to prove a quenched version of this limit.
Keywords
Boltzmann planar maps · local convergence A planar map M is a connected planar graph, possibly with loops and mul-tiple edges, together with an embedding into the plane. Usually one edge isdirected and distinguished as the root edge. Various analytic, combinatorial,and probabilistic techniques for studying models of random planar maps havebeen developed, see [3,6]. The bijection by [5] encodes planar maps as mobiles,which are vertex-labelled 4-type planar trees. This allows for a generating pro-cedure for certain models of random planar maps using 4-type Galton–Watsontrees, see [16]. For bipartite Boltzmann planar maps, a bijection constructed by[12] simplifies the generating procedure to use only monotype Galton–Watsontrees. However, it is an open problem whether a full reduction to mono-typetrees is possible in the non-bipartite case, hence the need to study multi-typeGalton–Watson trees for this purpose persists. Recent work by Stephenson [17] establishes local convergence of condi-tioned regular critical multi-type Galton–Watson trees, and applies this con-vergence to a conditioned Boltzmann planar map M n . The main application isa limit theorem that shows how an Infinite Boltzmann Planar Map ˆ M describesthe asymptotic behaviour of the vicinity of the root-edge of M n as n → ∞ . Benedikt StuflerVienna University of TechnologyInstitute of Discrete Mathematics and GeometryE-mail: The author thanks Sigurdur ¨Orn Stef´ansson for related comments. a r X i v : . [ m a t h . P R ] F e b Benedikt Stufler
This generalizes local convergence results for bipartite maps by [4] and specialcases like triangulations and quadrangulations by [2,14].The present work establishes a corresponding quenched version of the limittheorem. Roughly speaking, the difference is that instead of studying the prob-ability for the vicinity of the root-edge of M n to have a certain shape, weestablish laws of large numbers for the number of corners, faces, and verticeswhose vicinity has this shape. See Theorem 1. Our main tools are quenchedlimits of rerooted multitype trees established recently in [21]. As an application, we deduce quenched local convergence of the randomplanar map M tn with n edges and a positive weight t > M tn assumes a map M with n edges with probability proportional to t v( M ) ,with v( M ) denoting the number of vertices of M . See Theorem 2. The vertexweighted random planar map M tn is related to the study of uniform randomplanar graphs, see [11,7]. We apply the quenched local convergence of M tn inthe subsequent paper [19] to deduce local convergence of the uniform randomplanar graph. Notation
We let N = { , , , . . . } denote the collection of non-negative integers, and N the collection of positive integers. The law of a random variable X : Ω → S with values in some measurable space S is denoted by L ( X ). If Y : Ω → S (cid:48) isa random variable with values in some measurable space S (cid:48) , we let L ( X | Y )denote the conditional law of X given Y . All unspecified limits are taken as n → ∞ . Convergence in probability and distribution are denoted by p −→ and d −→ . We say an event holds with high probability if its probability tends to 1as n becomes large. For any sequence ( a n ) n ≥ of positive real numbers we let O p ( a n ) denote a random variable Z n such that ( Z n /a n ) n ≥ is stochasticallybounded. Index of terminology
The following list summarizes frequently used terminology. ξ An unordered D -offspring distribution ξ = ( ξ i ) i ∈ G , page 5. i ( · ) Number of vertices of type i ∈ G , page 4. | · | γ Sum of vertices weighted depending on their type, page 4 T ( η ) A ξ -Galton–Watson tree with (possilby random) root type η , page 6. T κ Like T ( κ ), but non-root vertices of type κ receive no off-spring, page 6. The results of the present work were initially part of [21]. The paper was split duringthe review process following a referee’s recommendation.uenched local convergence of Boltzmann planar maps 3 ˆ T κ A random tree with a marked leaf of type κ . Distributedlike T κ biased by the number of vertices with type κ ,page 6.ˆ T ( κ ) A random infinite tree with a marked vertex of type κ anda spine that grows backwards, page 6.ˆ T κ,ι A random tree with root type κ and a marked vertex oftype ι . Obtained by biasing T κ by the number of verticesof type ι , page 6.ˆ T ( κ, ι ) A random infinite tree with a marked vertex of type ι anda spine that grows backwards, page 6. M e denote the collection of finite planar maps with an oriented rootedge. The origin of this root edge is called the root vertex. The face to the leftof the oriented root-edge is called the root face. Likewise, we let M v denote thecollection of finite planar maps that only have a specified root vertex insteadof an oriented root edge. We also let M f denote the collection of finite planarmaps that only carry a marked root face instead. In the following, M refersto M e , M v , or M f , as all related concepts are analogous for these three cases.Note that M is countably infinite.For any integer k ≥ M k ⊂ M of planar mapswhere each vertex has “distance” at most k from the root. Here “distancefrom the root” refers to the graph distance from the root vertex in case ofvertex rooted maps, or the graph distances to the ends of the root-edge foredge rooted maps. For face rooted maps, we define the distance as the lengthof some shortest path from the vertex to the boundary of the marked face. Weequip M k with the discrete topology.The projection U k : M → M k maps a planar map M to the k -neighbourhood U k ( M ) of its root. Dependingon whether M refers to M e , M v , or M f we view U k ( M ) as equipped with anoriented root edge, a root vertex, or a root face.The local topology on the collection M is the coarsest topology that makesthese projections continuous. This projective limit topology is metrizable by d loc ( M, M (cid:48) ) = 11 + sup { k ≥ | U k ( M ) = U k ( M (cid:48) ) } , M, M (cid:48) ∈ M . Benedikt Stufler
The space ( M , d loc ) is not complete. One way to complete it, is to form thespace M of coherent sequences M = { ( M , M , . . . ) | U i ( M i +1 ) = M i for all i ≥ } ⊂ M N . We may interpret M as a subset of M , and extend d loc and U k ( · ) in a canonicalway. This makes M a Polish space, see [8, Prop. 1] for details.Let ( M n ) n ≥ be a sequence of random finite planar maps, and let u n beeither a uniformly selected vertex, oriented edge, or face. This makes the pair( M n , u n ) a random element of M . Here we forget about any possibly presentroot of M n , and only consider u n as the new root. Distributional convergence of( M n , u n ) n ≥ is equivalent to distributional convergence of each neighbourhood U k ( M n , u n ), k ≥
0, as n tends to infinity. If ˆ M is a random element of M , then( M n , u n ) d −→ ˆ M (1)is equivalent to P ( U k ( M n , u n ) = M ) → P ( U k ( ˆ M ) = M ) (2)for each fixed integer k ≥ M ∈ M as n → ∞ .Using language from statistical physics, this form of convergence is also called annealed convergence.The collection M ( M ) of probability measures on the Borel sigma algebraof M is a Polish space with respect to the weak convergence topology. Theconditional distribution L (( M n , u n ) | M n ) is a random element of M ( M ).We say ( M n , u n ) converges in the quenched sense, if the random probabilitymeasure L (( M n , u n ) | M n ) converges in distribution to a random element of M ( M ). For the special case where the limit is almost surely constant andgiven by the law L ( ˆ M ) of some random element ˆ M of M , we say ( M n , u n )converges in the quenched sense towards ˆ M .2.2 Limits of rerooted multi-type treesGiven an integer D ≥
1, a D -type plane tree T is a plane tree where we assign toeach vertex a type from { , . . . , D } . In particular, T has a root vertex, and foreach vertex we have a linear order on its collection of children. For 1 ≤ j ≤ D we let j T denote the total number of vertices with type j . Furthermore, forany vector γ = ( γ , . . . , γ D ) ∈ R D we set | T | γ = D (cid:88) i =1 γ i i T. (3)If we distinguish a vertex v of T , we form a marked tree ( T, v ). The subtreeconsisting of v and all its descendants is called the fringe subtree of T at v .For any integer k ≥ extended fringe subtree f [ k ] ( T, v )consisting of the fringe subtree of T at the k th ancestor of v , marked at the uenched local convergence of Boltzmann planar maps 5 vertex corresponding to v . Of course, this only makes sense if v has height atleast k in T . Otherwise, we set f [ k ] ( T, v ) to some placeholder value.The path from v to the root of T is called the spine of the marked tree( T, v ). We may also consider marked trees where this spine has a countablyinfinite length, such that v has a countably infinite number of ancestors. Welet X denote the collection of all finite marked D -type trees and all marked D -type trees with an infinite spine such that all extended fringe subtrees arefinite.The collection X may be endowed with a metric d X such that for all T • , T • ∈ X d X ( T • , T • ) = (cid:40) , f ( T • ) (cid:54) = f ( T • )2 − sup { k ≥ | f [ k ] ( T • )= f [ k ] ( T • ) } , f ( T • ) = f ( T • ) . (4)This makes ( X , d X ) a Polish space, see [21, Prop. 1].Let ( T n ) n ≥ be a sequence of random finite D -type trees. Let G ⊂ { , . . . , D } denote a non-empty subset, such that the probability for T n to have verticeswith type in G tends to 1 as n becomes large. Let v n be uniformly selectedamong all vertices of T n with type in G . Then ( T n , v n ) is a random elementof X . We say ( T n , v n ) convergences in the annealed sense towards a randomelement T • of X , if ( T n , v n ) d −→ T • in the usual sense of distributional convergence of random elements of thePolish space X . The conditional distribution L (( T n , v n ) | T n ) is a randomelement of the collection M ( X ) of Borel probability measures on X . That iswe take the tree T n (this is where the randomness comes from) and considerthe uniform distribution on all marked versions of T n where the marked vertexhas type in G . We say T • is the quenched limit of ( T n , v n ), if L (( T n , v n ) | T n ) d −→ L ( T • )in the sense of distributional convergence of random elements of the Polishspace M ( X ).2.3 Galton–Watson treesLet D ≥ D -type Galton–Watson tree is a random locallyfinite D -type plane tree defined as follows. Let ξ = ( ξ i ) ≤ i ≤ D be a family ofrandom elements ξ i ∈ N D . For any integer 1 ≤ κ ≤ D the ξ -Galton–Watson T ( κ ) starts with a single root vertex with type κ . For all 1 ≤ i ≤ D any vertexof type i receives offspring vertices according to an independent copy of ξ i ,with the j th coordinate (for 1 ≤ j ≤ D ) corresponding to the number of chil-dren with type j . For our purposes, we will always assume that the collection Benedikt Stufler of all children is ordered uniformly at random. If η is a random element of { , . . . , D } , independent from all previously considered random variables, welet T ( η ) denote the mixture of ( T ( κ )) ≤ κ ≤ D that assumes T ( κ ) with proba-bility P ( η = κ ) for each 1 ≤ κ ≤ D . That is, here the type of the root vertexis random and distributed like η .We define T κ similar to T ( κ ), only that non-root vertices with type κ receive no offspring. Let us assume that T κ is a.s. finite and E [ κ T κ ] = 2 . (5)This allows us to define the κ -biased version ˆ T κ with distribution P ( ˆ T κ = ( T κ , u )) = P ( T κ = T κ ) (6)for any pair ( T κ , u ) of a finite G -type tree T κ (with the root having type κ and all non-root vertices of type κ having no offspring) and a non-root leaf u of T κ with type κ .We construct a random tree ˆ T ( κ ) that has an infinite “backwards” growingspine u , u , . . . of type κ vertices, such that u (cid:96) +1 is an ancestor (not necessarilyparent) of u (cid:96) for all (cid:96) ≥
0. The construction is as follows. We start with thevertex u that becomes the root of an independent copy of T ( κ ). The vertex u becomes the root of an independent copy of ˆ T κ , which has a marked leaf.All non-marked leaves of type κ become roots of independent copies of T ( κ ),and we identify the marked leaf with u (“glueing” the two vertices together).We proceed in this way with an ancestor u of u and so on, yielding an infinitebackwards growing spine u , u , . . . of type κ vertices.The tree ˆ T ( κ ) constitutes the multi-type analogue of Aldous’ invariantsin-tree constructed in [1] for critical monotype Galton–Watson trees. Theabbreviation sin stands for single infinite path .Suppose that ι ∈ { , . . . , D } is a type. If the number of non-root type ι -vertices in T κ has a finite non-zero expectation E , we may form the ι -biasedversion T κ,ι of T κ with distribution P ( ˆ T κ,ι = ( T κ , u )) = P ( T κ = T κ ) /E (7)for any pair ( T κ , u ) of a finite G -type tree T κ (with the root having type κ and all non-root vertices of type κ having no offspring) and a non-root leaf u of T κ with type ι . This allows us to construct the tree ˆ T ( κ, ι ) analogous toˆ T ( κ ) with the only difference being that in the construction we start with atype ι vertex u that becomes the root of an independent copy of T ( ι ), andfor u we use T κ,ι instead of a copy of T κ . Hence u , u , . . . have type κ , but u has type ι . We recall important background on Boltzmann planar maps [16] and theBouttier–Di Francesco–Guitter transformation [5]. Our presentation follows uenched local convergence of Boltzmann planar maps 7 closely that of [17, Sec. 5], with some additional emphasis in Section 3.2 onhow the labels of a Boltzmann mobile may be constructed from conditionallyindependent choices for each vertex of the underlying Galton–Watson tree.3.1 The Boltzmann distribution on planar mapsThe collection of all finite planar maps with an oriented root edge and an ad-ditional marked vertex will be denoted by M . Throughout we let q = ( q n ) n ≥ denote a family of non-negative numbers such that q n > n ≥
3. To any element M ∈ M we assign a weight W q ( M ) = (cid:89) f q deg( f ) . (8)Here the index f ranges over the faces of the planar map M , and deg( f )denotes the degree of the face f . That is, deg( f ) is the number of half-edgeson the boundary of the face f . (The reason why we count half-edges insteadof edges is that an edge on the boundary has to be counted twice if both of itssides are incident to the face.) A weight-sequence q is said to be admissible , if Z q := (cid:88) M ∈M W q ( M ) < ∞ . (9)In this case, we may form the Boltzmann distributed (vertex marked) planarmap M with distribution given by P ( M = M ) = W q ( M ) /Z q , M ∈ M . (10)Likewise we may form analogously the Boltzmann planar map ˜ M (and condi-tioned versions thereof) by using the class of maps without a marked vertexinstead of M . Note that ˜ M and M follow different distributions, as M is biasedby the number of vertices.3.2 Mobiles obtained from branching processesA pointed map from M is said to be positive, neutral, or negative, if the originof the directed root edge is closer, equally far away, or farther away from themarked vertex than the destination of the root edge. We let M + , M , and M − denote the corresponding subclasses of M , and form the sums Z + q , Z q ,and Z − q as in (9), but with the sum index constrained to the correspondingsubclass. For all x, y ≥ f • ( x, y ) = (cid:88) k,k (cid:48) ≥ (cid:18) k + k (cid:48) + 1 k + 1 (cid:19)(cid:18) k + k (cid:48) k (cid:19) q k + k (cid:48) x k y k (cid:48) , (11) f (cid:5) ( x, y ) = (cid:88) k,k (cid:48) ≥ (cid:18) k + k (cid:48) k (cid:19)(cid:18) k + k (cid:48) k (cid:19) q k + k (cid:48) x k y k (cid:48) . (12) Benedikt Stufler
If the weight sequence q is admissible, we may define an irreducible 4-typeoffspring distribution ξ = ( ξ ) ≤ i ≤ as follows. Vertices of the first type producea geometric number of vertices of the third type: P ( ξ = (0 , , k, Z + q (cid:18) − Z + q (cid:19) k , k ≥ . (13)Vertices of the second type always produce a single offspring vertex of thefourth type, that is P ( ξ = (0 , , , . (14)Vertices of the third and fourth type only produce offspring of the first orsecond type. Their coordinates ξ , , ξ , and ξ , , ξ , are determined by E [ x ξ , y ξ , ] = f • ( xZ + q , y (cid:113) Z q ) f • ( Z + q , (cid:113) Z q ) (15) E [ x ξ , y ξ , ] = f (cid:5) ( xZ + q , y (cid:113) Z q ) f (cid:5) ( Z + q , (cid:113) Z q ) . (16)Here we have used that the denominators in (15) and (16) are finite. Thisfollows from [16, Prop. 1], see Section 3.4 below for details.For a type κ = 1 or κ = 2 we consider the following sampling procedure.The result is a random 4-type tree where the offspring is ordered and eachvertex v receives a label (cid:96) ( v ) with (cid:96) ( v ) ∈ Z if v has type 1 or 3, and (cid:96) ( v ) ∈ + Z otherwise.1. Consider the ξ -Galton–Watson tree T ( κ ) that starts with a single vertex oftype κ . We consider the offspring vertices as ordered in a uniformly selectedmanner.2. For each vertex v of type 3 or 4 in T ( κ ) with outdegree d ≥ v denote its parent and let v , . . . , v d denote its ordered offspring. For easeof notation, we set v d +1 := v . Note that v , . . . , v d all have types in { , } .Uniformly select a ( d + 1)-dimensional vector β T ( κ ) ( v ) = ( a , . . . , a d )satisfying the following two conditions:(a) (cid:80) di =0 a i = 0.(b) For all 0 ≤ i ≤ d :If v i and v i +1 both have type 1, then a i ∈ {− , , , . . . } .If v i and v i +1 both have type 2, then a i ∈ { , , , . . . } .If v i and v i +1 have different types, then a i ∈ {− / , / , / , . . . } .3. Assign to each vertex v ∈ T ( κ ) a label (cid:96) ( v ) in a unique way satisfying thefollowing conditions. uenched local convergence of Boltzmann planar maps 9 (a) The root of T ( κ ) receives label 0 if it has type 1 and label 1 / v of type 3 or 4 has offspring v , . . . , v d with d ≥ a , . . . , a d ) := β T ( κ ) ( v i ) and set (cid:96) ( v i ) := (cid:96) ( v ) + (cid:80) i − j =0 a j for all1 ≤ i ≤ d .This construction produces a so-called mobile . We emphasize that in thesecond step we choose for any vertex v of type 3 or 4 the vector β T ( κ ) ( v ) atrandom in a way that depends only on the ordered list of offspring vertices of v , their types, and the type of v (since it determines the type of its parent).In combinatorial language, ( T ( κ ) , β T ( κ ) ) is a special case of an multi-typeenriched plane tree . We refer to it as the canonical decoration of T ( κ ).3.3 The Bouttier–Di Francesco–Guitter transformationWe let T + denote an independent copy of ( T (1) , β T (1) ). We let T denote theresult of taking two independent copies of ( T (2) , β T (2) ) and identifying theirroots. Let ( T, β ) be a possible finite outcome of T + or T , and let ( (cid:96) ( v )) v ∈ T denote the corresponding labels. The Bouttier–Di Francesco–Guitter transfor-mation [5] associates a planar map Ψ ( T, β ) to the decorated tree (
T, β ) in sucha way that – the number of vertices of the map equals 1 + T , – the number of edges of the map equals T + T + T − – and the number of faces of the map equals T + T .The transformation Ψ is as follows. We draw T in the plane and orderthe corners according to the standard contour process that starts at the rootvertex. Let v , . . . , v p denote the ordered list of vertices of type 1 or 2 thatwe visit in the contour process. That is, a vertex gets visited multiple typesaccording to the number of angular sectors around it. We let (cid:96) , . . . , (cid:96) p denotetheir labels. We extend these lists cyclically, so that v ip + k = v k for i ≥ ≤ k ≤ p . We add an extra vertex r with type 1 outside of T and let its label (cid:96) ( r ) be one less than the minimum of labels of all type 1 vertices. For each1 ≤ i ≤ p we draw an arc between the vertex v i and its successor . If v i has type1 then the successor is the next corner in the cyclic list of type 1 with label (cid:96) i −
1. If there is no such corner, then we let r be the successor of v i . Likewise,if v i has type 2 then the successor of v i is the next corner of type 1 with label (cid:96) i − /
2, or r if there is no such corner. It is possible to draw all arcs so thatthey only may intersect at end points. We now delete the original edges of thetree T , as well as all vertices of type 3 and 4. Vertices of type 2 get erased aswell, merging the corresponding pairs of arcs. We are left with a planar maphaving a marked vertex r . If the root of T has type 1 we let the root edge bethe first arc that was drawn and have it point to the root of T . If the root of T has type 2 (and hence has precisely two children, both of type 4), we let theroot edge be the result of the merger of the two arcs incident to the root of T and let it point towards the successor of the first corner encountered in thecontour process. Figure 1 illustrates the transformation ψ for an example.The Boltzmann distributed map M is a mixture of the random maps M + , M , and M − obtained by conditioning M on belonging to M + , M , and M − .As observed by [16], it holds that Ψ ( T + ) d = M + and Ψ ( T ) d = M . Moreover, M − may be obtained from M + by reversing the direction of the root edge.3.4 Regimes of weight sequences[16, Prop. 1] showed that the weight sequence q is admissible if and only if thesystem of equations f • ( x, y ) = 1 − x (17) f (cid:5) ( x, y ) = y (18)has a solution ( x, y ) with x > x − xy ∂ x f (cid:5) ( x, y ) ∂ y f (cid:5) ( x, y ) 0 x x − ∂ x f • ( x, y ) xyx − ∂ y f • ( x, y ) 0 has spectral radius smaller or equal to one. Any such solution ( x, y ) necessarilysatisfies ( x, y ) = ( Z + q , (cid:113) Z q ) . (19)[16, Def. 1] termed an admissible weight sequence q critical , if the spectralradius of this matrix is equal to 1. This amounts to the condition x J f ( x, y ) + 1 = x ∂ x f • ( x, y ) + ∂ y f (cid:5) ( x, y ) , (20)with J f denoting the (signed) Jacobian of the function ( f • , f (cid:5) ) : R → ( R + ∪{∞} ) . It is termed regular critical , if additionally f • ( Z + q + (cid:15), y (cid:113) Z q + (cid:15) ) < ∞ (21)for some (cid:15) >
0. As was made explicit by [17], this applies to various usefulcases such as unrestricted maps or p -angulations for arbitrary p ≥
3. Theirreducible offspring distribution ξ is critical (or regular critical) if and only ifthe weight sequence q is critical (or regular critical). uenched local convergence of Boltzmann planar maps 11(a) A mobile from which we are going toconstruct a vertex-marked rooted planarmap. (b) Listing the corners incident to verticesof type 1 and 2. Adding a marked vertex.(c) Drawing arcs and distinguishing an ori-ented root edge. (d) Removing old edges, removing type 3and type 4 vertices, and replacing type 2vertices by arcs. Fig. 1: The correspondence between mobiles and vertex-marked rooted planarmaps.
Suppose that the weight-sequence q is regular critical. Let M n denote the q -Boltzmann planar map, conditioned on either having n vertices,or edges, or faces. Let u n denote either a uniformly selected vertex, half-edge,or face. There are integers a ≥ and d ≥ and a random infinite locallyfinite limit map ˆ M with finite face degrees such that, in the local topology forvertex-rooted or half-edge rooted or face-rooted planar maps, the conditionallaw L (( M n , u n ) | M n ) satisfies L (( M n , u n ) | M n ) p −→ L ( ˆ M ) (22) as n ∈ a + d Z tends to infinity. Of course, the limit object differs depending on which conditioning we chooseand which type of marking we select. The quenched limit (22) implies theannealed convergence ( M n , u n ) d −→ ˆ M (23)by dominated convergence. If u n denotes a uniformly selected half-edge, then(23) is the annealed convergence established by [17, Thm. 6.1] (see also [2,14,4,9,15]), who only required criticality in the case where M n is the Boltzmannmap with n vertices. [10] described a general method for deducing limits forthe vicinity of random vertices if a limit for the vicinity of a random cornersis known. The method applies to regular critical Boltzmann planar maps andother settings. Obtaining an explicit description of the limit was left as anopen question in [10], and the construction of the limit from an infinite mobilewith a backwards growing spine the proof of Theorem 1 resolves this questionin the present setting.Note that, as was shown by [17, Sec. 6.3.5], in the present setting thetotal variational distance between M n (a corner-rooted map with an additionalmarked vertex, not to be confused with u n ) and a q -Boltzmann map ˜ M n without a marked vertex tends to zero as n becomes large:lim n →∞ d TV ( M n , ˜ M n ) = 0 . (24)Hence Theorem 1 also holds for ˜ M n .4.1 Proof strategyThe existence of a ≥ d ≥ M n is well-defined for n ∈ a + d Z large enough, was shownby [17, Lem. 6.1]. Let γ ∈ N be either (1 , , ,
0) or (1 , , ,
1) or (0 , , , G = { } or G = { , , } or G = { , } accordingly. uenched local convergence of Boltzmann planar maps 13 Recall that T + denotes an independent copy of ( T (1) , β T (1) ), and T isthe result of taking two independent copies of ( T (2) , β T (2) ) and identifyingtheir roots. Recall also that the Boltzmann distributed map M is a mixture ofthe random maps Ψ ( T + ) d = M + , Ψ ( T ) d = M , and the result M − of reversingthe direction of the root-edge M + .Throughout the entire proof, a subscript n of a random tree denotes thatwe condition the tree on the event | · | γ = n if γ = (0 , , , | · | γ = n − γ = (1 , , , | · | γ = n + 1 if γ = (1 , , , n of a randommap will denote that we condition the map accordingly on having n faces orvertices or edges.Let κ ∈ { , . . . , } be a type. If we select a vertex v n from T n ( κ ) with typein G uniformly at random, then by [21, Thm. 6] L (( T n ( κ ) , v n ) | T n ( κ )) p −→ L ( ˆ T ( η )) (25)for a random type η that only depends on ξ and γ (and not on κ ). Addingcanonical decorations, this implies L (( T n ( κ ) , β T n ( κ ) , v n ) | T n ( κ )) p −→ L ( ˆ T ( η ) , β ˆ T ( η ) ) . (26)(See also [20] for a general theory of limits and fringe distributions of randomdecorated or enriched trees.)We are going to show that:a) The decorated tree ( ˆ T ( η ) , β ˆ T ( η ) ) corresponds to an infinite vertex-,corner-, or face-rooted map ˆ M via an extension of the Bouttier–DiFrancesco–Guitter transformation.b) The convergence (26) implies L (( M + n , u n ) | M + n ) p −→ L ( ˆ M ) . (27)c) Convergence of M − n follows from (27) and M n may be treated analo-gously as M + n .Having these intermediate results at hand, Theorem 1 immediately follows.In the following subsection, we verify the three claims individually.4.2 Claim a)In the third step of the procedure given in Section 3.2 we described a processfor transforming the decorations into labels. We cannot apply this processdirectly to ( ˆ T ( η ) , β ˆ T ( η ) ) since the tree has an infinite backwards growing spineof ancestors instead of a root. However, if we assign any valid label to a singlevertex v (with value in Z if v has type 1 or 3 and value in + Z if v hastype 2 or 4), then the decorations determine the labels of all other vertices.Moreover, the differences in the labels between any pair of vertices does notdepend on the label we started with. Hence let us assign a valid label 0 or 1 / to the marked vertex of ( ˆ T ( η ) , β ˆ T ( η ) ) (depending on whether its type η lies in { , } or { , } ), and extend this in a unique way according to the decorationsto labels ( (cid:96) ( v )) v ∈ ˆ T ( η ) . Lemma 1
The labels of the type ancestors of the marked vertex in ˆ T ( η ) have almost surely no lower bound.Proof First, let us observe that ˆ T ( η ) d = ˆ T (1 , η ) . (28)This could be verified directly, or as follows: The limit in Equation (25) is aspecial case of [21, Thm. 6], which was obtained as an application of the moregeneral theorem [21, Thm. 1]. We could just as well have applied [21, Thm.2, Rem. 2] instead, yielding that (25) holds with ˆ T (1 , η ) instead of ˆ T ( η ). Thisverifies (28).Let u , u , . . . denote the list of type 1 ancestors of the marked vertex inˆ T ( η ) (excluding the marked vertex itself, if it has type 1), so that u i +1 is anancestor of u i for all i ≥
1. Then the family of differences of labels (cid:96) ( u i +1 ) − (cid:96) ( u i ), i ≥ T , assigning labels accordinglywith an arbitrary starting value for the root of ˆ T , and forming the differenceof the labels between the root and the marked leaf of ˆ T . Thus, the labels( (cid:96) ( u i )) i ≥ form a random walk with i.i.d. steps and a random starting value (cid:96) ( u ).It is known that this random walk is centred: Indeed, consider the localweak limit ˜ T of T n (1) established by [17], that describes the asymptotic vicin-ity of the root (and not a random location) of T n (1). The construction of ˜ T isas follows. We start with a type 1 vertex that gets identified with an indepen-dent copy of ˆ T . All non-marked type 1 leaves become roots of independentcopies of T (1). For the marked leaf, we proceed recursively in the same wayas for the root (identifying it with the root of a fresh independent copy of ˆ T ,and so on). Hence T (1) has an infinite spine, obtained by concatenating inde-pendent copies of T (1). In particular, if we form the canonical decoration of ˜ T and assign labels accordingly (with, say, a starting value 0 for the root vertex),then the labels of the type 1 vertices of the spine form a random walk withi.i.d. steps and the same step distribution as for the random walk ( (cid:96) ( u i )) i ≥ .[17, Proof of Lem. 6.5] showed that this step distribution has average value 0.Hence, analogously as for [17, Lem. 6.5], it follows from [13, Thm. 9.2] thatalmost surely inf i ≥ (cid:96) ( u i ) = −∞ . This completes the proof.We may order the corners ( c i ) i ∈ Z incident to vertices of type 1 or 2 ofˆ T ( η ) such that for all i ∈ Z the corner c i +1 is the successor of c i in the clock-wise contour exploration. This allows us to canonically extend the Bouttier–Di uenched local convergence of Boltzmann planar maps 15 Francesco–Guitter transformation from Section 3.3 to assign an infinite locallyfinite planar map ˆ M to the infinite labelled tree ( ˆ T ( η ) , ( (cid:96) ( v )) v ∈ ˆ T ( η ) ). Here wedo not have to add an additional marked vertex, because the labels of type 1vertices along the backwards growing spine of ˆ T ( η ) have no lower bound. Byconstruction, all faces of ˆ M have finite degree.Depending on whether u n is a random vertex, half-edge, or face of M n , wemark ˆ M as follows. Let w denote the marked vertex of ˆ T ( η ), which has aninfinite number of ancestors. In the vertex case, w has type 1 and correspondscanonically to a vertex of ˆ M . We consider ˆ M as rooted at this vertex. In theface case, w has type 3 or 4 and corresponds canonically to a face. In thiscase, we consider ˆ M as rooted at this face. In the half-edge case, w has type1, 3, or 4 and corresponds canonically to an edge, which we orient accordingto an independent fair coin flip. In detail: If w has type 4, then it is the onlychild of a non-root type 2 vertex that corresponds to the edge obtained byjoining the arcs drawn at its two corners. Hence w corresponds canonically tothis edge. If w has type 1, then each of its corners corresponds to the edgewe drew when visiting this corner in the contour exploration. The number ofthese corners equals 1 plus the number of offspring vertices, all of which havetype 3. Hence w and its children correspond bijectively to the arcs we drewstarting at a corner of w . In particular, w corresponds canonically to an arc.Likewise, if w has type 3 it also corresponds canonically to an edge that wedrew starting at a corner of its type 1 parent.This verifies Claim a).4.3 Claim b)Suppose that κ = 1. The vertex v n of ( T n ( κ ) , β T n ( κ ) ) corresponds similarlyto a marked vertex or face or half-edge u (cid:48) n of M + n . Modifications in the corre-spondence may be required when v n or its parent is the root of T n ( κ ), but theprobability for this event tends to zero and hence we may safely ignore this.Furthermore, ( M + n , u n ) and ( M + n , u (cid:48) n ) may not follow the same distribution(for example, when u n is a uniform vertex, then u (cid:48) n is a uniform non-marked vertex, as u (cid:48) n is never equal to the additional vertex we added in the BDFGbijection). However, it is clear that there is an event (that depends on n ) whoseprobability tends to 1 as n becomes large, such that ( M + n , u n ) and ( M + n , u (cid:48) n )are identically distributed when conditioned on this event. Hence we may alsosafely ignore the difference between u n and u (cid:48) n . Using the continuous mappingtheorem, it hence follows from (26) that L (( M + n , u n ) | M + n ) p −→ L ( ˆ M ) . (29)This verifies Claim b). M − n , since the vicinityof a random point is not affected by the orientation of the root edge. As for M n , it follows from [17, Prop 2.2] that | T (2) | γ takes only values from a shiftedlattice, and has a density that varies regularly with index − / S (1) and S (2) of T (2) on the event | S (1) | γ + | S (2) | γ = n thenlim n →∞ min(( | S (1) | γ , | S (2) | γ ) | | S (1) | γ + | S (2) | γ = n ) d −→ | T (2) | γ . (30)This may easily be verified elementarily or be viewed as a special case forresults on general models of random partitions, see [18, Thm. 3.4, Prop 2.5].Consequently, all but a negligible number of vertices whose extended fringesubtree has a certain shape will lie in a giant component with size (“size”referring to | · | γ ) m − O p (1). If we let S denote the result of identifying theroots of S (1) and S (2) and let w n denote a uniformly selected vertex of theconditioned tree S n with type in G , then it follows by (25) that L (( S n , w n ) | S n ) p −→ L ( ˆ T ( η )) . (31)(Recall that above we assigned a clear meaning to all occurrences of n as asubscript of a random tree, making S n a conditioned version of S that dependson γ .) Hence, adding canonical decorations, L (( S n , β S n , w n ) | S n ) p −→ L ( ˆ T ( η ) , β ˆ T ( η ) ) . (32)Thus quenched convergence of M n towards ˆ M may be deduced in exactlythe same way using the mapping theorem as for M + n , only instead of usingEquation (26) we use Equation (32). This verifies Claim c). Let t > M tn denote a random planar map with n edgesthat assumes a map M (with n edges) with probability proportional to t v( M ) . Theorem 2
The random map M tn admits a distributional limit ˆ M t in the localtopology. Letting c n denote a uniformly selected corner of M tn , it holds that L (( M tn , c n ) | M tn ) p −→ L ( ˆ M t ) . (33) Proof
For any λ > q n = tλ n , n ≥ . (34)This way, any map with n edges and m faces receives weight λ n t m . We aregoing to argue below that for any t > λ so that q = ( q n ) n ≥ uenched local convergence of Boltzmann planar maps 17 is regular critical. By elementary identities of power series (compare with [17,Proof of Prop. 6.3]) the expressions in Equations (11) and (12) simplify to f • ( x, y ) = t (1 − Z )2 xZ , (35) f (cid:5) ( x, y ) = tλ (1 − λy ) Z , (36)with Z := (cid:115) − λ x (1 − λy ) . (37)Conditions (17) and (18) may be rephrased by Z = tt + 2 x − , (38)and Z = − λty ( λy − . (39)Note that this implies x >
1. Combining the last two equalities, we obtain λ = yt + 2 x + y − . (40)Plugging this expression into Equations (17) and (18) and noting that (38)implies x > y = √ x − √ t + x − √ x and λ = √ x − √ x √ t + x − t − x − t + 3 x + 1 and x > . (41)Moreover, for any triple ( x, y, λ ) of real numbers satisfying (41), we may easilyverify that Equations (17) and (18) hold (and that y > λ > x = 23 − t (cid:114) √ (cid:113) − ( t − t + 4( t − t + 4 (42)+ 12 (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) − t + 1)(2 t − t − (cid:114) √ − ( t − t / + ( t − t + 1 − √ (cid:113) − ( t − t + 89 ( t − + 8( t − . This solution is strictly bigger than 1 for any t > y and λ according to (41) we obtain a solution to Equations (17), (18), and (20).Hence for this choice of λ (depending on t ) the weight sequence q is critical.It is clear from the expressions (35), (36), (37) that q is even regular criticalin this case.Let M n denote the corresponding regular critical q -Boltzmann planar mapwith n edges. Let u n denote a uniformly selected corner of M n . As q is regular critical, it follows by Theorem 1 that there is an infinite random planar mapˆ M with finite face degrees such that L (( M n , u n ) | M n ) p −→ L ( ˆ M ) . (43)By (24), the q -Boltzmann map ˜ M n without a marked vertex consequentlysatisfies as well L (( ˜ M n , u n ) | ˜ M n ) p −→ L ( ˆ M ) . (44)The random planar map ˜ M n assumes any planar map M with n edges, m faces and k vertices with probability proportional to t m . That is, P ( ˜ M n = M ) = t m c n,t (45)for some constant c n,t > n and t . Euler’s formulaentails that m = 2 + n − k . Hence P ( ˜ M n = M ) = t − k c (cid:48) n,t , (46)with c (cid:48) n,t = c n,t t n again only depending on n and t . Thus,˜ M n d = M − tn . (47)Replacing t by t − , Equation (33) now follows from Equation (44). Acknowledgement
I warmly thank the associate editor and the referee for the thorough readingand helpful comments. In particular, for pointing out a simplification of theproof of Theorem 2.
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