Existence of gradient Gibbs measures on regular trees which are not translation invariant
EExistence of gradient Gibbs measures on regulartrees which are not translation invariant
Florian Henning ∗ †
Christof K¨ulske ∗ ‡
February 25, 2021
Abstract
We provide an existence theory for gradient Gibbs measures for Z -valued spin models on regular trees which are not invariant under transla-tions of the tree, assuming only summability of the transfer operator. Thegradient states we obtain are delocalized. The construction we providefor them starts from a two-layer hidden Markov model representationin a setup which is not invariant under tree-automorphisms, involvinginternal q -spin models. The proofs of existence and lack of translation in-variance of infinite-volume gradient states are based on properties of thelocal pseudo-unstable manifold of the corresponding discrete dynamicalsystems of these internal models, around the free state, at large q . Key words:
Gibbs measures, gradient Gibbs measures, regular tree,boundary law, heavy tails, stable manifold theorem.
The question whether statistical mechanics models with translation-invariantinteractions allow infinite-volume states which are not translation invariant hasa long history. A famous example which shows that this is possible are theDobrushin-states for the Ising model in zero external field on the integer lattice Z d , in dimensions d ≥
3. They can be obtained with plus/minus boundaryconditions on the upper/lower half of a sequence of cubes. At sufficiently lowtemperatures they break translation invariance, see [14] and also [6]. By con-trast, in low lattice dimensions d ≤ ∗ Ruhr-Universit¨at Bochum, Fakult¨at f¨ur Mathematik, D44801 Bochum, Germany † [email protected] ‡ [email protected], a r X i v : . [ m a t h . P R ] F e b e will be interested in the following in the case where the graph is a regulartree. Let us therefore start by recalling what is known for the ferromagneticIsing model in zero external field: In that case there are even uncountably manynon-automorphism invariant Gibbs measures in the full low-temperature region(Theorem 12.31 in [18]). This region is equivalently described as the region forwhich µ + (cid:54) = µ − where the latter states are the graph-automorphism invariantmeasures obtained as finite-volume limits with homogeneous plus respectivelyminus boundary conditions. Nonhomogeneous states of different type exist inregions of even lower temperatures, see [17] and also [25].In this paper we plan to investigate the analogous question for integer-valued gradient models on regular trees, which are described in terms of a nearestneighbor interaction potential U : Z → R which assigns the energetic contribution U ( ω x − ω y ) associated to an edge withendpoints x, y which is felt by an integer-valued spin configuration.For such gradient models we want to ask specifically for the possibility orimpossibility of existence of non-tree automorphism invariant gradient states .Note that gradient states, as distributions on increments of the variables,may exist in regimes where Gibbs states do not exist. This is the case in theclass of translation-invariant states for convex nearest neighbor potentials forreal variables on the two-dimensional integer lattice [16]. So, we stress that thequestion for non-automorphism invariant gradient states has to be distinguishedfrom the analogous question for non-automorphism invariant Gibbs states.For more on gradient models on the lattice with homogeneous interactionswe refer to [10], [13], [26], [2] and [4]. For more on gradient models on the latticein random environments, see [5], [15], [11],[12].What is known on trees for non-trivial automorphism-invariant gradientstates which do not arise as projections to the increments of the variables inany Gibbs measure? In recent work [20], [19] the existence of such states whichare different from the free state, was investigated. The authors found conditionson the transfer operator Q = e − βU , formulated in terms of smallness of suitable L p -norms, which in concrete modelscan be achieved by choosing the inverse temperature large. The existence proofwas then based on a contraction method. In the present paper we prove in full generality the following main result whichinformally reads:
Gradient models for any summable strictly positive transfer operator Q al-ways possess non-automorphism invariant gradient states, see Theorem 5. Westress that our theorem makes no assumption on convexity or monotonicity ofthe interaction, only summability and Q ( i ) > i ∈ Z . As there is alwaysthe free state of i.i.d increments, this implies that there is never uniqueness ofGGMs for any fixed summable Q , which seems surprising indeed at first look.How can one expect such a very general existence result to hold, and howto prove it, without any further properties of Q ?2 .2 Ideas of the proof In the first step one understands in Theorem 2 that GGMs can be obtainedas stochastic images of internal q -spin models which have a discrete rotationinvariance, also called clock-models, where q takes the meaning of a periodwhich may run through the integers 2 , , . . . .In this relation (possibly non-automorphism invariant) Gibbs measures ofthe q -spin models get mapped to GGMs for the gradient Hamiltonian. Thistype of relation was already exploited in [19] in the automorphism-invariantsetup, but will be shown to hold also in the non-automorphism invariant setup,by a new method of proof, which does not use invariance, see Section 3.3. Wedescribe in the following Subsection 3.5 why GGMs obtained as images in thisway are necessarily delocalized (i.e. they do not stem from a Gibbs measurewhich would also know absolute heights).In the second main part we investigate these internal q -spin models viaboundary law solutions, based on Zachary’s theory [28], at fixed q . We lookspecifically for solutions which are not translation invariant but radially sym-metric (around an arbitrary root). The corresponding boundary law formalismcan be naturally interpreted in terms of a q -indexed family of discrete-timedynamical systems S q : ∆ q → ∆ q on the simplex ∆ q , see the equations (23) and (24), where the non-linear map S q to be analyzed depends on the interaction Q and the period q . Backwards trajectories, stability analysis, and GGMs.
Non-automorphism-invariant states of the q -clock models are then obtained viatheir correspondence to backwards trajectories of the map S q . The difficultywith this argument is that such infinite backwards iterates may not alwaysexist, for arbitrary model parameters, periods, and initial values. This turnsthe existence problem for such GGMs in general highly nontrivial. Indeed,a general understanding of these dynamical systems, structures of fixed points,and their bifurcations in their dependence on Q , other than for very small valuesof q , poses challenging model-dependent tasks. For specific work on aspects ofinhomogeneity of solutions for Ising and Potts models, see [17], [24], [3], [23] and[27]. It turns out be very fruitful for our model-independent approach to focus onan important common property of the maps S q , shared by all gradient modelsfor summable Q , and periods q . Indeed, for any period q , in any parameterregime there is always the free state, which for any q corresponds to the trivialfixed point of S q provided by the equidistribution. The crucial point is, thatdepending on q , at fixed Q , the stability properties of this fixed point change, andthis has useful consequences for the existence of non-automorphism invariantstates. Namely, we show that for any summable Q , when the period q ≥ q ( Q )is large enough, then there is an unstable manifold of positive dimension ofthe map S q around the equidistribution. In this case non-trivial backwardstrajectories are obtained from starting points on this unstable manifold awayfrom the fixed point, and they yield the desired states.There is a part of the argument where we need care to be able to deal alsowith the exceptional cases of non-hyperbolicity (which may occur at exceptionalparameter values), and this is where we invoke the pseudo-unstable manifold3heorem of [7]. This yields a countable family of distinct non-trivial measuresindexed by q , see Proposition 4.By contrast to this general existence theorem, obtained for large enoughperiods q , the other interesting related question after existence of non-trivialsolutions at fixed periods q requires specific properties of the spectrum of thetransfer operator Q , see Theorem 4. For a quantitative discussion of this inthe context of the SOS-model as well as for a particular polynomially decayingtransfer operator, see Theorem 6 in Section 5. Proving lack of translation invariance of the associated GGMs.
To complete the proof of existence of GGMs which are not invariant undertranslations we need in a final step to understand how to get from a backwardstrajectory of S q to the associated GGM we are finally interested in. We show viaa local argument beyond linearization on the local pseudo-unstable manifold (seeSection 4.4) that the non-automorphism invariance of the radially symmetric q -b.l. solutions we construct really survives the map to the corresponding gradientstate. In fact, we obtain even non- translation-invariance when the Cayley treeis viewed as group acting on itself, which is stronger.The paper is organized as follows. Section 2 contains the definitions of themodel, and of the basic notions of Gibbs measures, gradient Gibbs measures,and tree-indexed Markov chains. Section 3 describes the map from period- q boundary laws to GGMs, allowing for inhomogeneity. Section 4 containsour existence results for non-invariant GGMs, which are explained in termsof backwards trajectories maps S q . Section 5 illustrates the theory for twoprototypical models.Finally, the proofs are given in Section 6. Acknowledgements
Florian Henning is partially supported by the Research Training Group 2131
High-dimensional phenomena in probability-Fluctuations and discontinuity ofGerman Research Council (DFG). 4
Definitions
We consider models on the Cayley-tree Γ d = ( V, L ) of order d ≥ Z as local state space and denote the set of height-configurations Z V by Ω. Let Z be equipped with the σ -algebra given by its power set and forany Λ ⊂ V let σ Λ : Z V → Z Λ , ( ω x ) x ∈ V (cid:55)→ ( ω x ) x ∈ Λ denote the coordinate spinprojection to the spins inside Λ. Then we consider the measurable space (Ω , F )where F := σ ( σ { x } | x ∈ V ) is the product- σ -algebra. For any subvolume Λ ⊂ V we denote by F Λ := σ ( σ { x } | x ∈ Λ) the σ -algebra on Ω generated by the spinsinside the volume Λ.The term Cayley tree (or d -regular tree) means a connected graph withoutcycles where each vertex has exactly d +1 nearest neighbors. We call two vertices x, y ∈ V nearest neighbors if they are connected by an edge b = { x, y } ∈ L .A collection of n edges { x, x } , { x , x } , . . . , { x n − , y } is called a path (oflength n ) from x to y , whereas an infinite collection of nearest neighbor pairswill be called an infinite path . For two vertices x, y the distance d ( x, y ) is definedas the length of the shortest path from x to y . Besides the set of unorientededges L which contains two-element subsets of V we also consider the set (cid:126)L of oriented edges which contains ordered pairs of vertices. If we restrict to aconnected subset Λ ⊂ V and set L Λ := {{ x, y } ∈ L | x, y ∈ Λ } then (Λ , L Λ ) isthe subtree of Γ with vertices inside Λ. Similarly, (cid:126)L Λ := { ( x, y ) ∈ (cid:126)L | x, y ∈ Λ } for any subset Λ ⊂ V .Furthermore, for any Λ ⊂ V we define its outer boundary by ∂ Λ := { x / ∈ Λ : d ( x, y ) = 1 for some y ∈ Λ } . If Λ ⊂ V is finite then we write Λ ⊂⊂ V . As outlined in Chapter 1.2 of [25],the Cayley tree Γ d = ( V, L ) of order d as a planar graph can be represented bythe free product G d of d + 1 cyclic groups of second order (i.e. groups whichcontain exactly two elements). Every element in G d is a finite word of symbolswhere each two adjacent symbols are from different groups.The group representation of V is then obtained as follows: Fix any root ρ ∈ V . Then ρ is represented by the unit e ∈ G d . The d + 1 nearest neigh-bors are enumerated counter-clockwise by the symbols a , . . . a d +1 . Now for any n ≥ v n be any vertex at distance n to the root. Then v n has a uniquenearest neighbor v n − lying on the shortest path from ρ to v n . v n − is repre-sented by a word of length n −
1. Let its rightmost symbol be a j . Then thegroup representation of v n is obtained by adding a symbol different from a j onthe right. Adding the symbol a j again one gets back to v n − . This enumerationof the d + 1 nearest neighbors to v n − in terms of the symbols a , . . . , a d +1 con-veniently corresponds to the counter-clockwise ordering in a planar embeddingof the graph. Two words are multiplied by concatenation and reduction.A useful concept in the context of tree-indexed Markov chains is the notionof future and past of a vertex which can be also found in chapter 12 of [18].Given any vertex v ∈ V , we write (cid:126)L v := { ( x, y ) ∈ (cid:126)L | d ( v, x ) = d ( v, y ) + 1 } (1)5or the set of edges pointing towards v and v (cid:126)L := { ( x, y ) ∈ (cid:126)L | d ( v, y ) = d ( x, v ) + 1 } (2)for the set of edges pointing away from v . Then] − ∞ , xy [ := { v ∈ V | ( x, y ) ∈ v (cid:126)L } (3)denotes the past of the oriented edge ( x, y ) ∈ (cid:126)L. Using this notation, a tree-indexed Markov chain on (Ω , F ) is a probability measure µ such that for any( x, y ) ∈ (cid:126)L µ ( σ ( x,y ) = i | F ] −∞ ,xy [ ) = µ ( σ ( x,y ) = i | F x ) µ − a.s. On the space of height-configurations we consider a symmetric nearest-neighborinteraction potential φ with corresponding transfer operator Q defined by Q b ( ζ ) := exp ( − φ b ( ζ ))for any edge b = { x, y } ∈ L and ζ ∈ Z b .The kernels of the Gibbsian specification ( γ Λ ) Λ ⊂⊂ V then read γ Λ ( σ Λ = ω Λ | ω ) = Z Λ ( ω ∂ Λ ) − (cid:89) b ∩ Λ (cid:54) = ∅ Q b ( ω b ) . (4)A Gibbs measure for the specification γ is a probability measure µ on (Ω , F )such that for all finite Λ ⊂ V and any A ∈ F µ ( A | F Λ c ) = γ Λ ( A | · ) µ -a.s. (5)This is equivalent to µγ Λ = µ for any finite Λ ⊂ V .We denote the set of Gibbs measures by G ( γ ).In this paper we focus on the special case of symmetric gradient interactions,i.e. for any edge b = { x, y } ∈ LQ b ( ω x , ω y ) = Q b ( ω x − ω y ) = exp( − βU b ( ω x − ω y )) , (6)where the parameter β > U b : Z → [0 , ∞ ) is a symmetric function.We are insterested in the particular case of measures which are invariant un-der joint translations of the local state space Z . Given any height-configuration ω = ( ω x ) x ∈ V we define the respective gradient configuration ∇ ω ∈ Z (cid:126)L by setting( ∇ ω ) ( x,y ) := ω y − ω x for any edge ( x, y ) ∈ (cid:126)L . Clearly,( ∇ ω ) ( x,y ) = − ( ∇ ω ) ( y,x ) for all ( x, y ) ∈ (cid:126)L. (7)In the other direction, from connectedness of the tree and absence of loops itfollows that a given gradient configuration satisfying the symmetry constraint(7) and prescription of the height at a fixed vertex defines a unique height-configuration. Hence the setΩ ∇ = { ( ζ ( x,y ) ) ( x,y ) ∈ (cid:126)L | ζ ( x,y ) = − ζ ( y,x ) for all ( x, y ) ∈ (cid:126)L }
6f gradient configurations bijectively corresponds to the quotient Z V / Z , the setof relative heights. For any subset Λ ⊂ V we denote by η Λ : Ω ∇ → Z (cid:126)L Λ thegradient spin projection to edges with both vertices inside the volume Λ. EquipΩ ∇ with the product- σ -algebra F ∇ = σ ( η ( x,y ) | ( x, y ) ∈ (cid:126)L ). For any Λ, we set F ∇ Λ = σ ( η ( x,y ) | ( x, y ) ∈ (cid:126)L Λ ). By construction, for any finite connected Λ ⊂ V the σ -algebra F ∇ Λ can be identified with the set of all events in F Λ which areinvariant under joint height-shift of all spins.In this paper we are interested in probability measures on the space of gra-dient configurations which are Gibbs in the sense that they are invariant underthe respective gradient configuration for the transfer operator Q . We note thatdue to the absence of cycles the complement of any (finite) subtree (Λ , L Λ ) of Γdecomposes into distinct connected components. This means that informationon the gradients outside of Λ does not determine a relative height-configurationon Λ c , by which we understand an element of Z Λ c / Z . Hence an event A ∈ F Λ c which is invariant under joint height-shift at all sites is in general not measur-able with respect to F ∇ Λ c . Therefore we will introduce a further outer σ -algebra T Λ which incorporates both the gradients outside the subtree (Λ , L Λ ) and therelative heights at the boundary, by which we understand an element of Z ∂ Λ / Z .More precisely, if we fix any (Λ , L Λ ), any vertex x ∈ ∂ Λ and any absolute height ω x ∈ Z then any gradient configuration ζ ∈ Ω ∇ gives rise to a unique heightconfiguration on Z ∂ Λ which depends only on the values of the gradient spinvariables inside Λ ∪ ∂ Λ. This follows from connectedness of the subtree (Λ , L Λ ).Hence we obtain an F ∇ Λ ∪ ∂ Λ -measurable function [ η ] Λ : Ω ∇ → Z ∂ Λ / Z . Here, theset Z ∂ Λ is endowed with the product- σ -algebra and Z ∂ Λ / Z is endowed with the σ -algebra generated by the projection. Then T Λ is given by T Λ = σ (( η ( x,y ) ) ( x,y ) ∈ (cid:126)L Λ c , [ η ] Λ ) . (8) In this section we give a general construction for probability measures on thegradient space which are invariant under the kernels of the gradient specifi-cation for a given gradient potential. These gradient Gibbs measures will beconstructed from (possibly spatially inhomogeneous) height-periodic functionssatisfying an appropriate version of Zachary’s [28] boundary law equation. Theresults of this section are a generalization of the homogeneous case consideredin [22] and [19] for which new proof methods become necessary.
The marginals of a Gibbs measure for a nearest-neighbor potential in a finitesubtree (Λ , L Λ ) can be written as the product of the associated transfer oper-ator evaluated at the spins at the edges with at least one vertex in the sub-tree and some F ∂ Λ -measurable function. By Zachary [28] this function canbe expressed by so-called boundary laws and one obtains a one-to-one relationbetween boundary laws and those Gibbs measures which are also tree-indexed7arkov chains, covering the class of extremal Gibbs measures. We cite the the-orem in its original form allowing also for nonhomogeneous transfer operators.We will use it later only for homogeneous interactions, but nonhomogeneousboundary law solutions. Definition 1.
A family of functions { λ xy } ( x,y ) ∈ (cid:126)L with λ xy := λ ( xy ) ∈ [0 , ∞ ) Z and λ xy (cid:54)≡ is called a boundary law for the family of transfer operators { Q b } b ∈ L ifi) for each ( x, y ) ∈ (cid:126)L there exists a constant c xy > such that the boundarylaw equation λ xy ( ω x ) = c xy (cid:89) z ∈ ∂x \{ y } (cid:88) ω z ∈ Z Q zx ( ω x , ω z ) λ zx ( ω z ) (9) holds for every ω x ∈ Z andii) for any x ∈ V the normalizability condition (cid:88) ω x ∈ Z (cid:16) (cid:89) z ∈ ∂x (cid:88) ω z ∈ Z Q zx ( ω x , ω z ) λ zx ( ω z ) (cid:17) < ∞ (10) holds true. Then the associated theorem reads
Theorem 1 (Theorem 3.2 in [28]) . Let ( Q b ) b ∈ L be any family of transfer oper-ators such that there is some ω ∈ Ω with Q { x,y } ( i, ω y ) > for all { x, y } ∈ L and any i ∈ Z . (11) Then for the Markov specification γ associated to ( Q b ) b ∈ L we have:i) Each boundary law ( λ xy ) ( x,y ) ∈ (cid:126)L for ( Q b ) b ∈ L defines a unique tree-indexedMarkov chain Gibbs measure µ ∈ G ( γ ) with marginals µ ( σ Λ ∪ ∂ Λ = ω Λ ∪ ∂ Λ ) = ( Z Λ ) − (cid:89) y ∈ ∂ Λ λ yy Λ ( ω y ) (cid:89) b ∩ Λ (cid:54) = ∅ Q b ( ω b ) , (12) for any connected set Λ ⊂⊂ V where y ∈ ∂ Λ , y Λ denotes the unique n.n. of y in Λ and Z Λ is the normalization constant which turns the r.h.s. intoa probability measure.ii) Conversely, every tree-indexed Markov chain Gibbs measure µ ∈ G ( γ ) ad-mits a representation of the form (12) in terms of a boundary law (uniqueup to a constant positive factor). In the context of gradient potentials 6, the requirement 11 means strictpositivity of the family ( Q b ) b ∈ L . One approach in reducing complexity of thesystem of equations (9) for gradient potentials 6 is assuming that all occurringfunctions λ xy ∈ [0 , ∞ ) Z are periodic functions on Z for some common period q ∈ { , , . . . } . 8 emark 1. In [28], boundary laws are formally defined as equivalence classesof families of functions, two functions being equivalent if and only if one isobtained by multiplying the other one by a suitable edge-dependent positive con-stant. Equivalent representatives of a boundary law are associated to the sametree-indexed Markov chain Gibbs measure. Hence we can impose a further con-straint to select a representative, e.g. by fixing the value of an arbitrary normto be one, as it will be done in the following.
In what follows we assume that the transfer operator { Q b } b ∈ L is summable, i.e. (cid:80) i ∈ Z | Q b ( i ) | < ∞ for all b ∈ L .We call a family ( λ qxy ) ( x,y ) ∈ (cid:126)L of functions λ qxy ∈ [0 , ∞ ) Z a q - height-periodicboundary law for the transfer operator { Q b } b ∈ L if it solves the boundary lawequation 9 and for any ( x, y ) ∈ (cid:126)L the function λ qxy : Z → (0 , ∞ ) is q -periodic.Clearly, there is a one-to-one correspondence between the set of q -height-periodicboundary laws for a transfer operator { Q b } b ∈ L and the set of boundary laws onthe finite local state space Z q = Z /q Z for the associated fuzzy transfer operator ( Q qb ) b ∈ L where Q qb (¯ i ) := (cid:80) j ∈ ¯ i = i + q Z Q b ( j ) , ¯ i ∈ Z q . One direction is simply givenby setting λ qxy (¯ i ) = λ qxy ( j ) for any j ∈ i + q Z . Hence, a q -height periodicboundary law (normalized to be an element of the unit simplex ∆ q ) can becomputed by solving the finite-dimensional system of equations λ qxy (¯ i ) = (cid:81) z ∈ ∂ { x }\ y (cid:80) ¯ j ∈ Z q Q q { x,z } (¯ i − ¯ j ) λ qzx (¯ j ) (cid:107) (cid:81) z ∈ ∂ { x }\ y (cid:80) ¯ j ∈ Z q Q q { x,z } (¯ · − ¯ j ) λ qzx (¯ j ) (cid:107) , ¯ i ∈ Z q . (13)at any edge ( xy ) ∈ (cid:126)L .Note that finiteness of all Q qb is equivalent to summability of all Q b whichexplains the necessity of this assumption. In what follows, we assume that q ∈ { , , . . . } and ( Q b ) b ∈ L is any symmetrictransfer operator of the form (4) such that for any b ∈ L the fuzzy transferoperator Q qb is strictly positive.Height-periodic boundary laws do not fulfil the normalizability condition(10). This means that the measure formally given by (12) is not defined. The q -spin fuzzy chain However, employing a two-step procedure by which we first sample a tree-indexed Markov chain, the so-called fuzzy chain on ( Z Vq , F q ) according to 12,where F q = ⊗ x ∈ V Z q is the product- σ -algebra generated by the fuzzy-spinprojections ¯ σ { x } : Z Vq → Z q , we are able to construct gradient measures on(Ω ∇ , F ∇ ) which satisfy a DLR-equation. The fuzzy chain itself is an elementof the set of Gibbs measures with respect to the fuzzy specification γ q whosekernels are given by γ q Λ (¯ σ Λ = ¯ ω Λ | ¯ ω ) = Z q Λ (¯ ω ∂ Λ ) − (cid:89) b ∩ Λ (cid:54) = ∅ Q qb (¯ ω b ) . (14)9or any finite Λ ⊂ V . After sampling a fuzzy chain, we independently applykernels to the edges each describing a distribution of total increments (as ele-ments of Z ) along an edge given the increment of the fuzzy chain (which is anelement of Z q ) along the respective edge. The so obtained probability measureon the space of gradients (Ω ∇ , F ∇ ) is thus a hidden Markov model where thetransition mechanism is not acting on the site-variables as it is more common,but acting on edge variables.We will first present the two steps of construction and then give the pre-cise definition of the respective DLR-equation and prove that the measure con-structed solves them. From q -spin fuzzy increments to Z -valued increments For any { x, y } ∈ L define a kernel ρ qQ { x,y } from Z q to 2 Z by setting: ρ qQ { x,y } ( j | ¯ s ) = χ ( j ∈ ¯ s ) Q { x,y } ( j ) Q q { x,y } (¯ s ) , (15)where χ denotes the indicator function.Then define a map T qQ : M ( Z Vq ) → M (Ω ∇ , F ∇ ) from spin- q measures onvertices to gradient measures by setting T qQ ( µ q )( η Λ = ζ Λ ) = (cid:88) ¯ ω Λ ∈ Z Λ q µ q (¯ σ Λ = ¯ ω Λ ) (cid:89) ( x,y ) ∈ w (cid:126)L, x,y ∈ Λ ρ qQ { x,y } ( ζ ( x,y ) | ¯ ω y − ¯ ω x )(16)where Λ ⊂ V is any finite connected set and w ∈ V is an arbitrary site. This de-scribes independent sampling over the edges with weight given by Q conditionalon the increment class.We note that due to the fact that ρ qQ { x,y } ( − j | − ¯ s ) = ρ qQ { x,y } ( j | ¯ s ) for all { x, y } ∈ L and any ¯ s ∈ Z q , j ∈ Z , the definition of the map T qQ does not dependon the concrete choice of the vertex w .Given any q -periodic boundary law ( λ qxy ) ( x,y ) ∈ (cid:126)L we will write ν λ q := T qQ ( µ λ q ) (17)where µ λ q ∈ G ( γ q ) is the fuzzy chain on Z Vq associated to ( λ qxy ) ( x,y ) ∈ (cid:126)L by The-orem 1.As the measures ν λ q will be of particular interest, we will write down afurther marginals representation. Lemma 1.
Let Γ( x, y ) ⊂ (cid:126)L denote the set of edges in the shortest path betweenvertices x and y and ¯ i ∈ Z q the mod- q projection of an integer i . Then thegradient measure ν λ q can be represented as ν λ q ( η Λ ∪ ∂ Λ = ζ Λ ∪ ∂ Λ ) = Z − (cid:88) ¯ s ∈ Z q (cid:89) y ∈ ∂ Λ λ qyy Λ (¯ s + (cid:88) b ∈ Γ( w,y ) ¯ ζ b ) (cid:89) b ∩ Λ (cid:54) = ∅ Q b ( ζ b ) , (18) where Λ ⊂ V is any finite connected volume, w ∈ Λ is any fixed site and Z Λ isa normalization constant.
10o to proofThe factor in parenthesis is the Radon-Nikodym derivative with respect tothe free measure and indicates dependence. Applying Lemma 1 to a singletonΛ = { x } for any x ∈ V and taking the boundary law equation (9) into accountthen gives in particular: Lemma 2.
Let ( x, y ) ∈ (cid:126)L be an oriented edge. Then we have for the single-edgemarginal of the gradient measure ν λ q ( η ( x,y ) = ζ ( x,y ) ) = Z − x,y ) Q { x,y } ( ζ ( x,y ) ) (cid:88) ¯ s ∈ Z q λ qxy (¯ s ) λ qyx (¯ s + ζ ( x,y ) ) where ζ ( x,y ) ∈ Z . Go to proofLemma 2 will be used later to prove lack of translation invariance.
First, we formally define a gradient specification γ (cid:48) . Afterwards, we will stateTheorem 2 on the gradient Gibbs property of the image of the map T qQ appliedto Gibbs measures on Z q . Definition 2.
Consider the outer σ -algebra T ∇ Λ (see (8) ) Then the gradientGibbs specification is defined as the family of probability kernels ( γ (cid:48) Λ ) Λ ⊂⊂ V from (Ω ∇ , T ∇ Λ ) to (Ω ∇ , F ∇ ) given by (cid:90) F ( ρ ) γ (cid:48) Λ ( d ρ | ζ ) = (cid:90) F ( ∇ ϕ ) γ Λ ( d ϕ | ω ) (19) for all bounded F ∇ -measurable functions F , where ω ∈ Ω is any height-configurationwith ∇ ω = ζ . Our first structural result is the following Theorem 2, which also covers thenon-homogeneous case.
Theorem 2.
The map T qQ maps Gibbs measures on Z Vq for the fuzzy specifica-tion γ q to gradient Gibbs measures for the gradient Gibbs specification γ (cid:48) . Go to proof
We show that the sum of increments of gradient Gibbs measures for spatially ho-mogeneous transfer operators along infinite paths diverge, and thus the heightsdelocalize. The following result is an extension of our result (Thm.4 in [19]) forthe special case of tree-automorphism gradient Gibbs measures to nonhomoge-neous GGMs.
Proposition 1. If ( λ qxy ) ( x,y ) ∈ (cid:126)L is a q -height-periodic boundary law solution fora spatially homogeneous family of positive transfer operators Q then the possiblynonhomogeneous gradient Gibbs measure ν λ q associated to it via (16) delocalizesin the sense that ν λ q ( W n = k ) n →∞ → for any total increment W n along a pathof length n and any k ∈ Z . Go to proof 11
Existence theory for non-invariant GGMs ofarbitrarily large periods q via pseudo-unstablemanifold theorem In this section we are investigating the possibility to apply backwards iterationof the boundary law equation (13) for radially symmetric (with respect to anarbitrary fixed root) q -height-periodic boundary laws to a spatially homogeneoustransfer operator Q . We will obtain the following surprising general result: Forany summable strictly positive spatially homogeneous Q , for large enough periodthere are always non-automorphism invariant GGMs. We stress that there isno assumption on the existence of different automorphism invariant GGMs. Assuming radial symmetry of the boundary law λ with respect to a vertex ρ ∈ V ,the boundary law equation (13) at any x ∈ ∂ { ρ } reads λ ρx = H q ( λ yρ ) := ( Q q λ yρ ) (cid:12) d (cid:107) ( Q q λ yρ ) (cid:12) d (cid:107) (20)where y ∈ ∂ { ρ } \ { x } . Here, we identified the fuzzy transfer operator Q q withthe symmetric circulant matrix ( Q q (¯ i − ¯ j )) ¯ i, ¯ j ∈ Z q and Q q λ yρ denotes the matrixproduct. For any v, w ∈ ∆ q and any s ∈ R , the vectors v (cid:12) s ∈ [0 , ∞ ) q \{ (0 , . . . , } and v (cid:12) w ∈ [0 , ∞ ) q \ { (0 , . . . , } are defined as( v (cid:12) s ) i = ( v i ) s and ( v (cid:12) w ) i = v i · w i , where i ∈ { , , . . . , q − } . We will refer to the conjunction (cid:12) as the
Hadamard-product . Note that bycontrast to the notation e.g. used in [24], in this paper the (cid:12) -conjunction doesnot necessarily give a normalized object.To remove the outer exponent d on the r.h.s. of (20), we may consider thetransformation u = G d ( λ ) := λ (cid:12) d (cid:107) λ (cid:12) d (cid:107) and define the composed map S q := G d ◦ H q ◦ G d from ∆ q to ∆ q . This means that for any u ∈ ∆ q S q ( u ) = Q q u (cid:12) d (cid:107) Q q (cid:107) (cid:107) u (cid:12) d (cid:107) . (21)To be able to apply backwards iteration to the map S q we need to know itsspectrum. The following proposition gives a full description of it in terms of theFourier-transformˆ Q : [ − π, π ) → R ; ˆ Q ( k ) = (cid:88) n ∈ Z Q ( n ) cos( nk ) (22)of Q . 12 roposition 2. The differential DS q [ eq ] : T ∆ q → T ∆ q of the map S q computedin the equidistribution eq = (1 /q, . . . , /q ) has the (cid:98) q (cid:99) eigenvalues d ˆ Q (2 π jq )ˆ Q (0) where j ∈ { , . . . , (cid:98) q (cid:99)} . Go to proofTaking into account continuity of the Fourier transform k (cid:55)→ ˆ Q ( k ) in thereal variable k at k = 0, Proposition 2 implies the following Lemma 3.
Fix any summable Q with strictly positive elements. For any degree d there is a finite minimal period q ( d ) such that for all q ≥ q ( d ) at least oneeigenvalue of the linearization DS q [ eq ] on T ∆ q is bigger than one in absolutevalue.More generally, for any degree d , and any finite dimension u there is aminimal period q ( d, u ) such that for all q ≥ q ( d, u ) at least u eigenvalues of DS q [ eq ] on T ∆ q are bigger than one in absolute value. We would now like to apply the radially symmetric backwards iteration onthe local unstable manifold at the equidistribution corresponding to the strictlyexpanding eigenvalues of D S q [ eq ]. There is the problem that we may encounteralso cases of neutral eigenvalues (see Figure 1 below) (cid:12)(cid:12)(cid:12) d ˆ Q (2 π jq )ˆ Q (0) (cid:12)(cid:12)(cid:12) = 1 . At such points the hyperbolicity of the fixed point fails, and the standard sta-ble manifold theorem for discrete-time dynamical systems for hyperbolic fixedpoints is not available in general.We can however bypass this difficulty by employing the τ -unstable manifoldtheorem (e.g. Thm. 1.2.2 in [7]) which covers the case where the spectrum ofD S q [ eq ] is off a circle of radius τ >
1. This was pointed out to us by AlbertoAbbondandolo.
Theorem 3 (Theorem 1.2.2 in [7] applied to the C ∞ -map S q ) . Assume there issome τ > such that the spectrum of D S q [ eq ] is the disjoint union of two sets σ ⊂ {| λ | < τ } and σ ⊂ {| λ | > τ } and let U ⊂ T eq ∆ q denote the D S q [ eq ] -stablesubspace such that the spectrum of the restriction D S q [ eq ] | U is σ .Then there exists a unique germ of a C ∞ -manifold W τ = W τ ( S q ) having thefollowing properties: • S q ( W τ ) ⊂ W τ , • T eq W τ = U , • The restriction S q | W τ is the germ of a C ∞ -diffeomorphism and • for any u ∈ W τ , the backwards iteration S − nq tends to the equidistributionnot slower than τ − n , where distance is measured in any local chart.A representative of W τ will called a local τ -unstable (or pseudo-unstable) mani-fold for the map D S q near the equidistribution. .5 1.0 1.5 2.0 2.5 3.0 k - Q ( k ) Q ( ) Figure 1: On the Cayley tree of order d = 5, period q = 16 and the inverse squaremodel (see Section 5 below) the non-hyperbolic case occurs at an exceptionalvalue of a = π ≈ . Q ( k ) / ˆ Q (0) at k = j π ,where j = 1 , . . . ,
8. The third bar from the left at k = π hits the upper dashedhorizontal line marking the threshold d = and hence represents a neutraleigenvalue of D S q [ eq ]. To satisfy the hypothesis of Theorem 4, we may put τ = and slightly shift the dashed lines away from the horizontal axis to thesolid horizontal lines at ± .Employing Proposition 2 to decide when it is possible to apply Theorem 3 toconstruct a radially symmetric boundary law solution via backwards iterationof S q and then going over from boundary laws to gradient Gibbs measures viathe Theorems 1 and 2 we arrive at the following Theorem 4. The details of theconstruction and the promised lack of translation invariance are given in thesubsections below. Theorem 4.
Fix any period q and any degree d ≥ . Suppose that there is alevel τ > for which the Fourier transform ˆ Q of the transfer operator Q satisfiesi) | ˆ Q (2 π jq ) | (cid:54) = τd ˆ Q (0) for all indices j ∈ { , . . . , q − } andii) the strict inequality | ˆ Q (2 π jq ) | > τd ˆ Q (0) is satisfied for some index j ∈{ , . . . , q − } .Then there are gradient Gibbs measures of period q which are not translationinvariant. They are constructed from the non-homogeneous radially symmetricboundary law solutions obtained from backwards iteration on the local τ -unstablemanifold W τ of the non-linear map S q around the equidistribution. Go to proofAs a direct corollary of Theorem 4 we obtain our main result.
Theorem 5.
For any summable Q and any degree d ≥ there is a finite period q ( d ) such that for all q ≥ q ( d ) there are invariant gradient Gibbs measures ofperiod q which are not translation invariant. Remark 2.
We may compare this result to the different existence Theorem in[19] for automorphism-invariant states different from the free state. That theo-rem naturally works in a “low temperature regime” formulated in terms of cer-tain p -norms of Q , and provides non-uniqueness of automorphism-invariant tates for large enough periods q . Our present result is for states which are prov-ably not invariant under translations. It does not need any such low temperatureassumption and works for all summable Q . In the following three subsections we assume q ≥ q ( d ) such that existence of τ -unstable manifold W τ of the non-linear map S q (see (21)) around the equidis-tribution is given. In this subsection we explicitly construct a radially symmetric boundary lawwhich is not translation invariant via backwards iteration.Let u = ( u , . . . , u q ) be any starting value chosen from the local unstablemanifold W τ of the non-linear map S q (see (21)) around the equidistribution.Fix any vertex ρ ∈ V which we will refer to as the root .First we define the boundary law values at edges ( x, y ) ∈ (cid:126)L ρ pointing towards ρ , i.e. d ( ρ, x ) = d ( ρ, y ) + 1 = n for some n ∈ N by setting λ uxy = G d ( S − nq ( u )) = ( S − nq ( u )) (cid:12) d (cid:107) ( S − nq ( u )) (cid:12) d (cid:107) . (23)Here, S − nq denotes the n − S − q . By radial symmetryof the construction and definition of the function S q , the boundary law equationis solved at any such edge ( x, y ) pointing towards ρ . By construction theyconverge to the trivial fixed one as the distance goes to infinity.For edges pointing away from the root the recursion formula for the boundarylaw reads as follows: Lemma 4.
Consider an infinite path { ρ = x , x } , { x , x } , . . . where d ( x n , ρ ) = n . Then the recursion formula for the boundary law values at edges pointingaway from the root reads as follows: λ uρx = S q ( u ) (cid:12) d (cid:107) S q ( u ) (cid:12) d (cid:107) λ ux n x n +1 = Q q ( λ uxn − xn ) (cid:12) ( S − nq ( u )) (cid:12) d − (cid:107) Q q ( λ uxn − xn ) (cid:12) ( S − nq ( u )) (cid:12) d − (cid:107) , n ∈ N (24)Go to proof By definition of the local unstable manifold the boundary law values (23) atedges pointing towards the root ρ converge to the equidistribution as the dis-tance to the root tends to infinity. The recursion formula (24) describing theboundary law values at edges pointing away from the root is more complicated.Nonetheless we will show that they also converge to the equidistribution as thedistance to the root tends to infinity, a result which will then be employed toprove the lack of translation invariance of the associated gradient states statedin the Theorems 4 and 5.Define a n := S − n ( u ) (cid:12) d − . We know that for u in W τ we have a n → eq (cid:12) d − = C ( q, d ) eq . 15t is convenient to discuss (24) in terms of the following map F a : ∆ q → ∆ q where F a ( z ) = Q q z (cid:12) a (cid:107) Q q z (cid:12) a (cid:107) . Then the convergence result reads
Lemma 5.
Suppose that a n are defined as above and z := S ( u ) (cid:12) d (cid:107) S ( u ) (cid:12) d (cid:107) , for u ∈ W τ chosen in a sufficiently small neighborhood of the equidistribution eq in W τ .Then we have the convergence result F a n F a n − . . . F a ( z ) n →∞ → eq . This meansthat the boundary law functions λ ux n x n +1 converge to the equidistribution alongany infinite path pointing away from the root. Go to proof
In this subsection we will show that the gradient Gibbs measures to the bound-ary law constructed from a starting value u ∈ W τ is not translation invariantprovided u is chosen from some uncountable subset of W τ . Proposition 3.
Fix q ≥ q ( d ) . Then there exist uncountably many startingvalues u such that for any fixed ρ ∈ V the gradient Gibbs measure ν λ u forthe boundary law λ u with values given by the equations (23) and (24) is nottranslation invariant. We will now present the main ideas of the proof of Proposition 3 in termsof the following two Lemmas. To improve readability, we will use the followingnotation: Let T ¯ j : R Z q → R Z q ; ( T ¯ j w )( · ) := w ( · + ¯ j )denote the cyclic shift of the index by ¯ j ∈ Z q .Further let (cid:104)· , ·(cid:105) denote the Euclidean scalar product in R q .We will first compare the marginals’ distribution of the gradient Gibbs mea-sure ν λ q along an edge starting from the root ρ with the respective marginalalong an edge at far distance from ρ . Employing the convergence result for theboundary laws stated in Lemma 5 above then gives the following Lemma. Lemma 6.
Let u ∈ W τ be any starting value for the recursion (24) . If there issome ¯ j ∈ Z q such that (cid:104) Q q ( u (cid:12) d ) (cid:12) d , ( T ¯ j u ) (cid:12) d − u (cid:12) d (cid:105) (cid:54) = 0 (25) then the gradient Gibbs measure ν λ u is not translation invariant. Go to proof In the next step we may represent the ( C ∞ ) local τ -unstablemanifold W τ in a neighborhood of the equidistribution by the graph of a C ∞ -function defined on a neighborhood of the equidistribution in the tangent spaceT eq W τ . For more details see also the proof of Thm.1.4.1 in [7] where existenceof such a map is already shown to construct the local unstable manifold. Per-forming a second-order Taylor expansion then yields the third statement of thefollowing Lemma. 16 emma 7.
1. We may parametrize any element u ∈ W τ near the equidistribution interms of v ∈ T eq W τ in the corresponding stable linear space, in the form u ( v ) = eq + v + h ( v ) with h ( v ) ∈ T eq ∆ q in the orthogonal space to T eq W τ inT eq ∆ q , describing the deviation of the unstable manifold from its tangentspace.2. By Theorem 3 we have T eq W τ = T + eq W τ ⊕ T − eq W τ where T +( − ) eq W τ = span { w ∈ T eq ∆ q | w is eigenvector of D S q to λ > τ ( λ < − τ ) } .
3. Assume that T + , ( − ) eq W τ has positive dimension. Then there is an openneighborhood V +( − ) of ∈ T + , ( − ) eq W τ such that max ¯ j ∈ Z q |(cid:104) Q q ( u ( v ) (cid:12) d ) (cid:12) d , ( T ¯ j u ( v )) (cid:12) d − u ( v ) (cid:12) d (cid:105)| ≥ C (cid:107) v (cid:107) (26) holds for all v ∈ V +( − ) . Go to proof q Proposition 4.
Assume that s and t are coprime natural numbers. Then anyof the s -height-periodic gradient Gibbs measures constructed in the Theorems4 and 5 is different from all of the t -height-periodic gradient Gibbs measuresconstructed in these Theorems. Go to proof
Remark 3.
In particular, q -height-periodic GGMs indexed by distinct primesare distinct. In this section we will apply the general results on existence of translation non-invariant gradient Gibbs measures of general period q stated in Section 4 to twoconcrete examples. The first one is the well known SOS-model parametrizedby the inverse temperature β >
0. The second one, which we will refer to asinverse square model, is described by a transfer operator fixed to the value 1at zero and polynomially decaying of second order with linear dependence on aparameter a > a amount tohigher suppression of increments.The inverse square model serves as a manageable example in which thedifferential D S q [ eq ] can obtain both positive and negative eigenvalues. It is alsothe basis of Figure 1 presented in Section 4.Both models allow for explicit analysis, see Theorem 6. This works partic-ularly well in the two cases, as the Fourier transform of the transfer operator17as good monotonicity properties, and in particular an expression for its point-wise inverse in terms of explicit functions. For any parametrized model withsummable transfer operator p (cid:55)→ Q p the same can be done in principle, if oneprovides the necessary additional (possibly numerical) input for the discussionof the pointwise inverse of its Fourier transform k (cid:55)→ ˆ Q p ( k ). Model SOS Inverse square Q ( i ) exp( − β | i | ) χ ( i = 0) + χ ( i (cid:54) = 0) aj ˆ Q ( k ) e β − e β − e β cos k +1 a (3 k − πk + 2 π ) ˆ Q ( π )ˆ Q (0) tanh( β ) (1 − a π ) / (1 + a π ) Q q (¯ i ) cosh ( β ( i − q ) ) sinh( β q ) aq (cid:0) ζ (2 , iq ) + ζ (2 , − iq ) (cid:1) +1 + χ (¯ i (cid:54) = ¯0)( aq i − Figure 2: The two models parametrized by β ( a , respectively), their Fourier-transforms and their mod- q fuzzy operators. For ¯ i ∈ Z q , the representative i ∈ Z occuring in the last row of the table is taken from { , . . . , q − } . Here ζ ( s, w ) = (cid:80) ∞ n =0 1( n + w ) s denotes the Hurwitz zeta function which is accessible vianumerical methods. It is defined for w, s ∈ C where | s | > < R ( w ) ≤ q -state clock models, the existence criterion presented in Theorem 4does not explicitly refer to the fuzzy transfer operators. Nonetheless the fuzzytransfer operators are included in the synoptic Figure 2 to give the reader anidea of their appearance and also highlight the benefits of presenting Theorem4 in terms of (the Fourier transform of) the original Z -state transfer operator.The spectra of D S q [ eq ] (up to the factor d ) for particular choices of theparameters β and a are presented in Figure 3. β = β = β = k Q ( k ) Q ( ) (a) SOS-model a = = = k - - Q ( k ) Q ( ) (b) Inverse square-model Figure 3: The graphs of the function ˆ Q ( · ) / ˆ Q (0) (see Theorem 4) for the twomodels at different parameter values. 18 emark 4. Recall that by the Bochner-Herglotz representation theorem (eg.Thm 15.29 in [21]) the transfer operator Q as a function on the integers ispositive semidefinite (which by definition is equivalent to the positive semidef-initeness of all matrices of the form ( Q ( i a − i b )) ≤ a,b ≤ n for all integers n andall choices of i a ∈ Z ) if and only if it is the Fourier transform of a positivemeasure. In particular if Q (0) < Q ( i ) for some i (cid:54) = 0 then Q is not positivesemidefinite and hence ˆ Q must obtain negative values in parts of its domain.The inverse square-model provides an example for this phenomenon. Both models presented above have a decreasing Fourier transform at any par-ticular choice of parameters β and a . The following theorem gives explicit andoptimal parameter regions where Theorem 4 is applicable for all q ≥
2. More-over, in the spirit of Theorem 5, the minimal periods for the existence of gradientGibbs measures which are not translation invariant are presented as functionson the whole parameter domain.
Theorem 6.
Consider the Cayley tree of order d ≥ .1. We have the following parameter regions for which q -height periodic gradi-ent Gibbs measures which are not translation invariant exist for all periods q ≥ . (cid:40) β > arcosh( d +1 d − ) for the SOS-model a ∈ (0 , ∞ ) \ [ π d − d +2 , π d +1 d − ] Inverse square model .
2. Conversely, for < β ≤ arcosh( d +1 d − ) and a ∈ [ π d − d +2 , π d +1 d − ] , we havethe following minimal periods such that for all q greater or equal to themexistence of q -height periodic gradient Gibbs measures which are not trans-lation invariant is guaranteed. q SOS ( β, d ) = (cid:100) π arccos( d − ( d −
1) cosh( β )) (cid:101) for the SOS-model q InvSq ( a, d ) = (cid:100) ππ − (cid:113) π (1+ d ) − a (1 − d ) (cid:101) Inverse square modelHere (cid:100)·(cid:101) denotes the smallest integer bounding the nonnegative argumentfrom above.3. If a ∈ ( π d +1 d − , π d +1 d − ] then also -height-periodic gradient Gibbs measureswhich are not translation invariant exist for the Inverse square modelAber. Go to proof
Remark 5.
In the situation of the first statement of Theorem 6 with β > arcosh( d +1 d − ) and a < π d − d +2 at any q ≥ all eigenvalues of D S q [ eq ] are strictlygreater than meaning that the unstable manifold W τ ( S q ) ⊂ ∆ q has the fulldimension q − . Hence, the set of initial values in the construction of thegradient Gibbs measures lacking translation invariance has maximal degrees offreedom.By contrast, in the complement of these parameter regions, for large q someeigenvalues will fail to be greater than one in modulus. Hence, the unstablemanifold does not possess full dimensionality in that case. Proofs
Proof of Lemma 1.
Let Λ ⊂ V be any finite connected volume and w ∈ Λ anyfixed site. Then for any ζ Λ ∪ ∂ Λ ∈ Z Λ ∪ ∂ Λ by the equations (16) and (15) we have ν λ q ( η Λ ∪ ∂ Λ = ζ Λ ∪ ∂ Λ )= T qQ ( µ λ q )( η Λ ∪ ∂ Λ = ζ Λ ∪ ∂ Λ )= (cid:88) ¯ ω Λ ∪ ∂ Λ ∈ Z Λ ∪ ∂ Λ q µ λ q (¯ σ Λ ∪ ∂ Λ = ¯ ω Λ ∪ ∂ Λ ) (cid:89) ( x,y ) ∈ w (cid:126)L, x,y ∈ Λ ∪ ∂ Λ ρ qQ { x,y } ( ζ ( x,y ) | ¯ ω y − ¯ ω x )= (cid:88) ¯ ω Λ ∪ ∂ Λ ∈ Z Λ ∪ ∂ Λ q µ λ q (¯ σ Λ ∪ ∂ Λ = ¯ ω Λ ∪ ∂ Λ ) (cid:89) ( x,y ) ∈ w (cid:126)L, x,y ∈ Λ ∪ ∂ Λ χ (¯ ω y − ¯ ω x = ¯ ζ ( x,y ) ) Q { x,y } ( ζ ( x,y ) ) Q q { x,y } (¯ ζ ( x,y ) )(27)As Λ is connected, given the gradient configuration ζ Λ ∪ ∂ Λ ∈ Z Λ ∪ ∂ Λ thereis a one-to-one relation between the height ¯ ω w at site w and the set of heightconfigurations ¯ ω Λ ∪ ∂ Λ ∈ Z Λ ∪ ∂ Λ q for which all indicators in (27) are nonvanishing.Hence we can write the last expression in (27) as= (cid:88) ¯ s ∈ Z q µ λ q (¯ σ w = ¯ s, ¯( ∇ σ ) Λ ∪ ∂ Λ = ¯ ζ Λ ∪ ∂ Λ ) (cid:89) { x,y }∈ L, x,y ∈ Λ ∪ ∂ Λ Q { x,y } ( ζ ( x,y ) ) Q q { x,y } (¯ ζ ( x,y ) ) . (28)Now, by Theorem 1 applied to the finite-state-space measure µ λ q , at any¯ s ∈ Z q we have µ λ q (¯ σ w = ¯ s, ¯( ∇ σ ) Λ ∪ ∂ Λ = ¯ ζ Λ ∪ ∂ Λ )= Z − (cid:89) y ∈ ∂ Λ λ qyy Λ (¯ s + (cid:88) b ∈ Γ( w,y Λ ) ¯ ζ b ) (cid:89) b ∩ Λ (cid:54) = ∅ Q qb (¯ ζ b ) , (29)where Z Λ is a normalization constant. Inserting this into (28) finishes the proofof Lemma 1. Proof of Lemma 2.
Apply Lemma 1 to Λ = { x } . Then summing over ζ ( x,z ) for z ∈ ∂ { x } \ { y } gives ν λ q ( η ( x,y ) = ζ ( x,y ) ) = Z − (cid:88) ¯ s ∈ Z q λ qyx (¯ s + ¯ ζ ( x,y ) ) Q { x,y } ( ζ ( x,y ) ) × (cid:89) z ∈ ∂ { x }\{ y } (cid:88) ζ ( x,z ) λ qzx (¯ s + ¯ ζ ( x,z ) ) Q { x,z } ( ζ ( x,z ) ) (30)By the boundary law equation (9) the last expression in parentheses equals λ qxy (¯ s ) up to a positive constant. This finishes the proof of Lemma 2. Proof of Theorem 2.
By linearity of the DLR-equation (19) and the map T qQ ,and by extremal decomposition of Gibbs measures (e.g. Thm. 7.26 in [18]) itsuffices to prove the statement for extremal Gibbs measures of γ q . All of these20re tree-indexed Markov chains (see Theorem 12.6 in [18]). Hence by Theorem1 it suffices to show that all measures of the form ν λ q as defined in (17) areinvariant under the kernels 4.This means that we have to check ν λ q ( A | T ∇ Λ ) = γ (cid:48) Λ ( A | · ) ν λ q -a.s. for all A ∈ F ∇ and all finite subtrees Λ ⊂ V . Let Λ ⊂ Λ ∪ ∂ Λ ⊂ ∆ be any finitesubtrees and ω, ζ ∈ F ∇ any gradient configurations with ζ V \ Λ = ω V \ Λ . To easenotation, in what follows we will omit the projection mappings and simply write ν λ q ( ζ Λ ∪ ∂ Λ ) instead of ν λ q ( η Λ ∪ ∂ Λ = ζ Λ ∪ ∂ Λ ).Then we have ν λ q ( ζ Λ ∪ ∂ Λ | [ ω ] ∂ Λ , ω (∆ ∪ ∂ ∆) \ Λ ) ν λ q ( ω Λ ∪ ∂ Λ | [ ω ] ∂ Λ , ω (∆ ∪ ∂ ∆) \ Λ ) = 1 [ ζ ] ∂ Λ =[ ω ] ∂ Λ ν λ q ( ζ Λ ∪ ∂ Λ , ω (∆ ∪ ∂ ∆) \ Λ ) ν λ q ( ω Λ ∪ ∂ Λ , ω (∆ ∪ ∂ ∆) \ Λ ) , (31)where we used that [ ζ ] ∂ Λ is F ∇ Λ ∪ ∂ Λ measurable. By Lemma 1 it follows ν λ q ( ζ Λ ∪ ∂ Λ , ω (∆ ∪ ∂ ∆) \ Λ ) = (cid:88) ¯ s ∈ Z q (cid:89) y ∈ ∂ ∆ λ qyy ∆ (¯ s + (cid:88) b ∈ Γ( v,y ) ζ b ) (cid:89) b ∩ ∆ (cid:54) = ∅ Q b ( ζ b ) . By the assumption ζ V \ Λ = ω V \ Λ all factors in (31) depending only on verticesin Λ c cancel:1 [ ζ ] ∂ Λ =[ ω ] ∂ Λ ν λ q ( ζ Λ ∪ ∂ Λ | [ ω ] ∂ Λ , ω (∆ ∪ ∂ ∆) \ Λ ) ν λ q ( ω Λ ∪ ∂ Λ | [ ω ] ∂ Λ , ω (∆ ∪ ∂ ∆) \ Λ )= 1 [ ζ ] ∂ Λ =[ ω ] ∂ Λ (cid:81) b ∩ Λ (cid:54) = ∅ Q b ( ζ b ) (cid:80) ¯ s ∈ Z q (cid:81) y ∈ ∂ ∆ λ qyy ∆ (¯ s + (cid:80) b ∈ Γ( v,y ) ζ b ) (cid:81) b ∩ Λ (cid:54) = ∅ Q b ( ω b ) (cid:80) ¯ s ∈ Z q (cid:81) y ∈ ∂ ∆ λ qyy ∆ (¯ s + (cid:80) b ∈ Γ( v,y ) ω b ) . Now note that1 [ ζ ] ∂ Λ =[ ω ] ∂ Λ (cid:88) ¯ s ∈ Z q (cid:89) y ∈ ∂ ∆ λ qyy ∆ (¯ s + (cid:88) b ∈ Γ( v,y ) ζ b ) = 1 [ ζ ] ∂ Λ =[ ω ] ∂ Λ (cid:88) ¯ s ∈ Z q (cid:89) y ∈ ∂ ∆ λ qyy ∆ (¯ s + (cid:88) b ∈ Γ( v,y ) ω b ) , as the outer summation is done over the same terms with shifted indices if therelative heights at ∂ Λ coincide. This finally gives ν λ q ( ζ Λ ∪ ∂ Λ | [ ω ] ∂ Λ , ω (∆ ∪ ∂ ∆) \ Λ ) ν λ q ( ω Λ ∪ ∂ Λ | [ ω ] ∂ Λ , ω (∆ ∪ ∂ ∆) \ Λ ) = 1 [ ζ ] ∂ Λ =[ ω ] ∂ Λ (cid:81) b ∩ Λ (cid:54) = ∅ Q b ( ζ b ) (cid:81) b ∩ Λ (cid:54) = ∅ Q b ( ω b )= γ (cid:48) Λ ∪ ∂ Λ ( ζ Λ ∪ ∂ Λ | ω ) γ (cid:48) Λ ∪ ∂ Λ ( ω Λ ∪ ∂ Λ | ω ) . Summing over all possible ζ Λ ∪ ∂ Λ ∈ F ∇ Λ ∪ ∂ Λ and taking the limit ∆ → V (thespecification γ (cid:48) is quasilocal) concludes the proof. Proof of Proposition 1.
Consider a path ( b , b , . . . , b n ) of length n and let η b := σ x − σ y denote the gradient spin variable along the edge b = ( x, y ) ∈ (cid:126)L and set W n := (cid:80) ni =1 η b i . Finally let ¯ σ = (¯ σ x i ) i =1 ,...,n +1 be the fuzzy chain on Z q alongthis path.For any fixed k ∈ Z we have ν λ q ( W n = k ) = (cid:90) µ λ q (d¯ σ ) ν λ q ( W n = k | ¯ σ ) .
21n the rest of the proof we will show that ν λ q ( W n = k | ¯ σ ) n →∞ → σ . To start with, first recall that ν λ q ( W n = k | ¯ σ ) is just the productmeasure of the measures ρ Qq describing the marginals along the edges of thepath conditioned on the increment of the fuzzy chain along the respective edge.Given the increment of the fuzzy chain along an edge, the measure ρ Qq doesnot have any spatial dependence. Hence the remainder of the proof is a directgeneralization of the proof of Theorem 4 in [19] which states delocalization of ν λ q in the special case of a spatially homogeneous boundary law λ q . Proof of Proposition 2 .
The proof consists of first proving thatD S q [ eq ] = d (cid:107) Q (cid:107) Q q (32)and afterwards expressing the spectrum of Q q by the Fourier-transform of Q .For any v ∈ ∆ q we have S q ( v ) = Q q v (cid:12) d (cid:107) Q q (cid:107) (cid:107) v (cid:12) d (cid:107) = (cid:107) Q q (cid:107) Q q ( G d ( v )). Directcalculation gives the following Lemma 8.
For any s (cid:54) = 0 the map G s : ∆ q → ∆ q ; z (cid:55)→ z (cid:12) s (cid:107) z (cid:12) s (cid:107) is a C ∞ -diffeomorphism with inverse G s . The differential at z ∈ ∆ q is given byD G s [ z ]( v ) = s (cid:107) z (cid:12) s (cid:107) (cid:18) z (cid:12) s − (cid:12) v − z (cid:12) s (cid:104) z (cid:12) s − , v (cid:105)(cid:107) z (cid:12) s (cid:107) (cid:19) for any v ∈ T z ∆ q . Hence, applying the chain rule we arrive atD S q [ eq ]( v ) = 1 (cid:107) Q q (cid:107) Q q (D G d [ eq ]( v ))= 1 (cid:107) Q q (cid:107) dq d − Q q ( eq (cid:12) d − (cid:12) v ))= d (cid:107) Q q (cid:107) Q q ( v )= d (cid:107) Q (cid:107) Q q ( v )where the second term of D G d [ eq ]( v ) vanishes as (cid:104) eq (cid:12) d − , v (cid:105) = 0.The eigenvalues of the q -circulant matrix Q q are given by Fourier transform λ j ( Q q ) = q − (cid:88) r =0 Q q ( r ) e i πjrq (33)where j = 0 , , . . . , q −
1. The corresponding eigenvectors are √ q (1 , ω j , . . . , ω q − j )with ω j = e i πjq .By symmetry of Q q we are reduced to the real cosine-Fourier modes andonly 1 + (cid:98) q (cid:99) distinct eigenvalues for which we can choose an orthogonal basisof eigenvectors in R q . These are given by suitable linear combinations of the22eal and imaginary parts of the complex eigenvectors stated above. Insertingthe definition of the fuzzy operator function in terms of sums of elements ofequivalence classes we obtain λ j ( Q q ) = Q q (0) + q − (cid:88) r =1 Q q ( r ) cos (cid:18) πjrq (cid:19) = (cid:88) n ∈ Z Q ( n ) cos (cid:0) πn jq (cid:1) (34)where j = 0 , , . . . , (cid:98) q (cid:99) .Hence, we see that the eigenvalues of the q -fuzzy operator are given by theFourier transform of the original transfer operator[ − π, π ) (cid:51) k (cid:55)→ ˆ Q ( k ) = (cid:88) n ∈ Z Q ( n ) cos( nk ) (35)sampled at the finitely many values 2 π jq , j = 0 , , . . . , (cid:98) q (cid:99) .In the last step we insert (cid:107) Q (cid:107) = ˆ Q (0) into (32) finishing the proof ofProposition 2. Proof of Lemma 4.
Consider an infinite path { ρ = x , x } , { x , x } . . . .For any n ∈ N let y , . . . y d ∈ ∂ { x n } \ { x n +1 } such that y d = x n − . Thenthe set of oriented edges of the form ( y i , x n ), i = 1 , . . . d splits up into the edge( x n − , x n ) pointing away from the root and the d − y i , x n ), i = 1 , . . . d − y i , ρ ) = d( x n , ρ ) + 1 = n + 1. From radialsymmetry of the construction and the boundary law equation (13) it thus followsthat λ ux n x n +1 = Q q ( λ x n − x n ) (cid:12) ( Q q ( λ y x n )) (cid:12) d − (cid:107) Q q ( λ x n − x n ) (cid:12) ( Q q ( λ y x n )) (cid:12) d − (cid:107) = Q q ( λ x n − x n ) (cid:12) S q ( G d ( λ y x n )) (cid:12) d − (cid:107) Q q ( λ x n − x n ) (cid:12) S q ( G d ( λ y x n )) (cid:12) d − (cid:107) = Q q ( λ x n − x n ) (cid:12) ( S q ( S − nq ( u ))) (cid:12) d − (cid:107) Q q ( λ x n − x n ) (cid:12) ( S q ( S − nq ( u ))) (cid:12) d − (cid:107) = Q q ( λ x n − x n ) (cid:12) ( S − nq ( u )) (cid:12) d − (cid:107) Q q ( λ x n − x n ) (cid:12) ( S − nq ( u )) (cid:12) d − (cid:107) where as above the (cid:12) -operator denotes the Hadamard product.In the case n = 0, i.e. for the edge ( ρ, x ), we simply obtain λ uρx = G d ( S q ( u )) = S q ( u ) (cid:12) d (cid:107) S q ( u ) (cid:12) d (cid:107) as all d edges of the form ( y i , ρ ), y , . . . , y d ∈ ∂ { ρ } \ { x } are pointing towardsthe root. Proof of Lemma 5.
It is convenient for the proof to work with distance on ∆ q provided by the restriction of the Euclidean norm.The proof rests on the following local uniform contraction estimate.23 emma 9. There is a positive contraction coefficient γ < , and there areneighborhoods U, W of the equidistribution such that for all a ∈ U and for all w, w (cid:48) ∈ W τ the following locally uniform contraction holds in two-norm. (cid:107) F a ( w ) − F a ( w (cid:48) ) (cid:107) ≤ γ (cid:107) w − w (cid:48) (cid:107) Proof of Lemma 9.
Choose γ strictly between M := max j =1 ,...,q − | λ j | and 1,where λ j denote the eigenvalues of the symmetric normalized transfer operator Q q (cid:107) Q q (cid:107) acting on the tangent space T eq ∆ q . We have for z (cid:55)→ F eq ( z ) = Q q z (cid:107) Q q z (cid:107) that the k -th component of the differential D F eq [ eq ] taken in z = eq is given byD F eq [ eq ] k = Q qk (cid:107) Q q (cid:107) where the denominator is the one-norm of the function Q q . Hence D F eq [ eq ] | T ∆ q = Q q (cid:107) Q q (cid:107) . Hence we know that the l -operator norm of D F eq [ eq ] | T eq ∆ q is equal to M <
1. But as the function ( z, a ) (cid:55)→ D F a [ z ] is jointly continuous, we mayperturb this estimate and see the following. For each γ strictly bigger than M ,we may find neighborhoods U, W of the equidistribution such that for all a ∈ U and for all z ∈ W the differential satisfies the uniform norm-estimate (cid:107) D F a [ z ] (cid:107) Op,l (T eq ∆ q ) ≤ γ But from this follows the desired Lipschitz property for the two-norm on T eq ∆ q .This concludes the proof of the lemma.We continue with the proof of Lemma 5 and split the relevant quantity intwo terms (cid:107) F a n F a n − . . . F a ( z ) − eq (cid:107) ≤ (cid:107) F a n F a n − . . . F a ( z ) − F a n F a n − . . . F a ( eq ) (cid:107) + (cid:107) F a n F a n − . . . F a ( eq ) − eq (cid:107) (36)To be able to apply the local uniform contraction property of Lemma 9, we firsthave to ensure that u can be chosen such that for all n ∈ N both F a n F a n − . . . F a ( eq )and F a n F a n − . . . F a ( z ) stay sufficiently close to the equidistribution. Afterwardswe show that both terms converge to zero as n tends to infinity. We start withestimating the second term of (36).Let ε >
0. We will show by induction on n ∈ N that (cid:107) F a n F a n − . . . F a ( eq ) − eq (cid:107) < ε for all n ∈ N . As ( a n ) n ∈ N converges exponentially fast to the equidis-tribution (since a ∈ W τ ), continuity of F in a allows to take the starting value a = a such that b n := (cid:107) F a n ( eq ) − eq (cid:107) < (1 − γ ) ε for all n ∈ N . In particular,this already proves the initial step. The induction step then follows from (cid:107) F a n F a n − . . . F a ( eq ) − eq (cid:107) ≤ (cid:107) F a n F a n − . . . F a ( eq ) − F a n ( eq ) (cid:107) + (cid:107) F a n ( eq ) − eq (cid:107) ≤ γ (cid:107) F a n − . . . F a ( eq ) − eq (cid:107) + b n , (37)where in the second inequality we used the induction hypothesis for n − (cid:107) F a n F a n − . . . F a ( eq ) − eq (cid:107) ≤ γ (cid:107) F a n − . . . F a ( eq ) − eq (cid:107) + b n ≤ · · · ≤ n (cid:88) j =0 γ n − j b j . (38)Fix ˜ ε >
0. As b n converges to zero as n tends to infinity, there is an m suchthat b j ≤ ˜ ε for all j ≥ m . Then n (cid:88) j =0 γ n − j b j = m − (cid:88) j =0 γ n − j b j + n (cid:88) j = m γ n − j b j ≤ γ n − m +1 − γ sup j ≥ b j + ˜ ε − γ . (39)Hence, considering ˜ ε ↓
0, we have (cid:107) F a n F a n − . . . F a ( eq ) − eq (cid:107) n →∞ → (cid:107) F a n F a n − . . . F a ( z ) − F a n F a n − . . . F a ( eq ) (cid:107) ≤ γ (cid:107) F a n − . . . F a ( z ) − F a n − . . . F a ( eq ) (cid:107) ≤ · · · ≤ γ n (cid:107) z − eq (cid:107) n →∞ → . (40)Here, the condition that F a n F a n − . . . F a ( z ) stays sufficiently close to the equidis-tribution again follows from induction on n provided that z is taken sufficientlyclose to the equidistribution. This concludes the proof of Lemma 5. Proof of Lemma 6.
Consider an infinite path { ρ = x , x } , { x , x } , . . . where d ( x n , ρ ) = n . Further assume that in the group representation x n = g . . . g n we have g (cid:54) = g m for all 1 < m ≤ n . For any n ∈ N define a translationΘ n : V → V ; Θ n ( · ) := g . . . g n · where g . . . g n is the group representation ofthe vertex x n . Then, for all m ∈ N , we have d ( ρ, Θ n ( x m )) = n + m . Employingthe fact that by Lemma 5 the boundary law functions λ ux n x n +1 and λ ρ,ux n +1 ,x n converge to the equidistribution as n tends to infinity, we will compare themarginals’ representation of the measure ν λ q along the edge { ρ, x } to thatalong the edge { x n x n +1 } = { Θ n ( ρ ) , Θ n ( x ) } for large n . The statement of thelemma then follows by showing that (25) implies that the marginals distributionalong the edge { ρ, x } is different from the equidistribution. Take any j , j ∈ Z .Then translation invariance of the measure ν λ u would imply that ν λ u ( η ( ρ,x ) = j ) ν λ u ( η ( ρ,x ) = j ) = ν λ u ( η ( x n ,x n +1 ) = j ) ν λ u ( η ( x n ,x n +1 ) = j ) (41)Inserting the statement of Lemma 2 for the marginal along an edge ν λ u ( η ( x n ,x n +1 ) = j ) = Z − x n ,x n +1 ) Q ( j ) (cid:88) ¯ i ∈ Z q λ ux n x n +1 (¯ i + ¯ j ) λ ux n +1 x n (¯ i )= Z − x n ,x n +1 ) Q ( j ) (cid:104) λ ux n x n +1 , T ¯ j ( λ ux n +1 x n ) (cid:105) n tend to infinity where the r.h.s converges to 1, we arriveat the necessary criterion (cid:104) λ uρ,x , T ¯ j λ ux ,ρ (cid:105)(cid:104) λ uρ,x , T ¯ j λ ux ,ρ (cid:105) = 1 . Hence, inserting λ ux ρ = u (cid:12) d (cid:107) u (cid:12) d (cid:107) and λ uρx = S ( u ) (cid:12) d (cid:107) S ( u ) (cid:12) d (cid:107) = Q q ( u (cid:12) d ) (cid:107) Q q ( u (cid:12) d ) (cid:107) the measure ν λ u is proven to be not invariant under Θ n if there are ¯ j , ¯ j ∈ Z q such that (cid:104) Q ( u (cid:12) d ) (cid:12) d , T ¯ j ( u ) (cid:12) d (cid:105) (cid:54) = (cid:104) Q ( u (cid:12) d ) (cid:12) d , T ¯ j ( u ) (cid:12) d (cid:105) . (42)In this case we may assume that ¯ j = ¯0 finishing the proof of the Lemma 6. Proof of Lemma 7. As W τ is a C ∞ -manifold, there is a C ∞ -function h defined ona neighborhood V ⊂ T eq W τ of (cid:126) u ( v ) := eq + v + h ( v )maps V onto some neighborhood U ⊂ W τ of eq . Moreover, the differential D h [ (cid:126) v (cid:55)→ (cid:104) Q ( u ( v ) (cid:12) d ) (cid:12) d , ( T ¯ j u ( v )) (cid:12) d − u ( v ) (cid:12) d (cid:105) around (cid:126) ∈ T eq W τ .First note that for any vector w ∈ R q and (cid:126) (cid:0) , , . . . (cid:1) T ∈ R q wehave (cid:104) (cid:126) , T ¯ j w − w (cid:105) = 0 for all ¯ j ∈ Z q . Hence, both the constant term and thefirst-order term in the expansion will vanish and only the mixed term in thesecond derivative remains. We haveD( u (cid:12) d )[ (cid:126) v ) = dq − d ( v + D h [ (cid:126) v )) = dq − d v and D( Q q ( u (cid:12) d ) (cid:12) d )[ (cid:126) v ) = dQ q ( eq (cid:12) d ) (cid:12) d − (cid:12) Q q (D( u (cid:12) d )[ (cid:126) v ))= d q − d (cid:107) Q (cid:107) Q q ( v ) . Thus, on the neighborhood V ⊂ T eq W τ of eq it follows (cid:104) Q q ( u ( v ) (cid:12) d ) (cid:12) d , ( T ¯ j u ( v )) (cid:12) d − u ( v ) (cid:12) d (cid:105) = 12 D ( (cid:104) Q q ( u ( v ) (cid:12) d ) (cid:12) d , ( T ¯ j u ( v )) (cid:12) d − u ( v ) (cid:12) d (cid:105) )[ (cid:126) v, v ) + O ( (cid:107) v (cid:107) )= 12 2 (cid:104) D( Q q ( u (cid:12) d ) (cid:12) d )[ (cid:126) v ) , D( T ¯ j u (cid:12) d )[ (cid:126) v ) − D( u (cid:12) d )[ (cid:126) v ) (cid:105) + O ( (cid:107) v (cid:107) )= d q − d ( d +1) (cid:107) Q (cid:107) (cid:104) Q q ( v ) , T ¯ j v − v (cid:105) + O ( (cid:107) v (cid:107) ) . (43)To estimate (cid:104) Q q ( v ) , T ¯ j v − v (cid:105) , we may sum over all ¯ j ∈ Z q . Then (cid:88) ¯ j ∈ Z q (cid:104) Q q ( v ) , T ¯ j v − v (cid:105) = − q (cid:104) Q q ( v ) , v (cid:105) , where the first term vanishes as v ∈ T eq ∆ q .Hence max ¯ j ∈ Z q |(cid:104) Q q ( v ) , T j v − v (cid:105)| ≥ |(cid:104) Q q ( v ) , v (cid:105)| . (44)26ow assume that T +( − ) eq W τ has positive dimension. Then v ∈ V ∩ T +( − ) eq W τ implies that the r.h.s of (44) is bounded from below by τ (cid:107) v (cid:107) . Combining (43)and (44) thus shows (26). This conludes the proof of Lemma 7. Proof of Proposition 4.
Let λ s ∈ Z (cid:126)Ls and l t ∈ Z (cid:126)Lt denote boundary laws obeyingthe recursions (23) and (24) constructed via backwards iteration on the respec-tive local unstable manifolds such that the condition of Lemma 7 is fulfilled. Let q = st . Define ˜ λ q ∈ [0 , ∞ ) Z q (cid:126)L and ˜ l q ∈ [0 , ∞ ) Z q (cid:126)L as the q -periodic continuationsof λ s and l t , i.e. at any edge ( x, y ) ∈ (cid:126)L the vector ˜ λ qxy ∈ [0 , ∞ ) Z q is defined asthe periodic continuation of the vector λ sxy ∈ [0 , ∞ ) Z s and similar for the vector˜ l q . By construction it follows that ˜ λ q and ˜ l q are boundary laws for the fuzzytransfer operator Q q . Moreover, ν ˜ λ q = ν λ s and ν ˜ l q = ν l t . Let x ∈ ∂ { ρ } . If ν ˜ λ q = ν ˜ l q then from the marginals’ representation of Lemma 2 at the edge { ρ, x } we obtain (cid:104) ˜ λ qxρ , T ¯ j (˜ λ qρx ) (cid:105) = C (cid:104) ˜ l qxρ , T ¯ j (˜ l qρx ) (cid:105) (45)for any j ∈ Z where the constant C > j and ¯ j ∈ Z q denotesthe mod- q projection. As ˜ λ q is the periodic continuation of an s -periodic vector,the l.h.s. of (45) is an s -periodic function in j ∈ Z . Similarly, the r.h.s is a t -periodic function in j . Now s and t were assumed to be coprime, hence the l.h.sis a constant function in j . In particular, for the boundary law λ s we necessarilyhave (cid:104) ˜ λ sxρ , T ¯ j (˜ λ sρx ) − ˜ λ sρx (cid:105) = 0at any ¯ j ∈ Z s which is excluded by the Lemma 7. Hence the measures ν λ s and ν l t must have different marginals along the edge { ρ, x } which concludes theproof of Proposition 4. Proof of Theorem 4.
Recall that the map S q describes the boundary law equa-tion (13) under the assumption of radial symmetry.If the condition 4 holds true then Proposition 2 describing the spectrum ofD S q [ eq ] and Theorem 3 ensure existence of a local unstable manifold W τ for S q near the equidistribution eq . As described in Section 4.2, for any starting value u ∈ W τ we then obtain a radially symmetric nonhomogeneous boundary lawsolution via backwards iteration of the map S q .By Theorem 1 this boundary law solutions corresponds to a Z q -valued Markovchain Gibbs measure which by Theorem 2 can be mapped to an integer valuedgradient Gibbs measure.At last, Proposition 3 guarantees that lack of invariance under translationsof the tree of the constructed boundary law solution carries over to the so-obtained gradient Gibbs measure for uncountably many choices of the startingvalue u ∈ W τ .This concludes the proof of Theorem 4. Calculation of Figure 2. OS-model:
The calculation of the Fourier transform is elementary as it onlyinvolves geometric series. The value of the fraction ˆ Q SOS ,β ( π ) / ˆ Q SOS ,β (0)then follows immediately. In the next step, we calculate the fuzzy transferoperator Q q SOS ,β . Let ¯ i ∈ Z q and i ∈ { , . . . , q − } ∩ ¯ i . Then we have Q q SOS ,β (¯ i ) = (cid:88) j ∈ Z exp( − β | i + qj | ) = cosh( β ( i − q ))sinh( β q )where in the last step we have used decomposition into geometric series. Inverse-square model:
To verify the expression for the Fourier-transform ofthe inverse-square transfer operator we compute its backwards Fourier-transform. For any nonzero integer j we have the elementary integralrelation (cid:82) π cos( kj )(3 k − πk + 2 π )d k = (6 π ) /j which is obtained frompartially integrating twice. From this the result follows.It remains to calculate the fuzzy transfer operator Q q ISq ,a .Let ¯ i ∈ Z q and i ∈ { , . . . , q − } ∩ ¯ i . Then we have Q q ISq ,a (¯ i ) = (cid:88) j ∈ Z Q ISq ,a ( i + qj )= Q ISq ,a ( i ) + βq ∞ (cid:88) j =1 j + iq ) + βq ∞ (cid:88) j =1 j − iq ) which proves the claim. Proof of Theorem 6.
SOS-model:
For the SOS-model at any β > Q SOS ,β ( k )ˆ Q SOS ,β (0) = e β − e β + 1 e β − e β cos k + 1 (46)which is strictly decreasing with k on [0 , π ]. Moreover, the expression isstrictly positive.Now the condition ˆ Q SOS ,β ( k ) > (1 /d ) ˆ Q SOS ,β (0) of Theorem 4 is equivalentto cos k > d − ( d −
1) cosh( β ) . (47)From the equation (47) we obtain the following:First, inserting the value k = π we arrive at the lower bound β =arcosh( d +1 d − ) for existence of non-invariant gradient Gibbs measures (n-t.i. GGM) of all periods q .Second, assuming β below this threshold and substituting k = πq , weobtain the minimal period q ( β, d ) for the SOS-model.28 nverse-square model: On the other hand, for the inverse square model atparameter a > Q ISq ,a ( k )ˆ Q ISq ,a (0) = a ( k − π ) + 1 − aπ aπ , (48)which is strictly decreasing with k on [0 , π ] and has a zero if and only if a ≥ π .Depending on the value of the expression (48) at k = π there are threedifferent regions for the parameter a :If a < π d − d +2 then ˆ Q ISq ,a ( π )ˆ Q ISq ,a (0) > d . Hence, by monotonicity ˆ Q ISq ,a ( k )ˆ Q ISq ,a (0) > d forall k meaning that existence of n-t.i. GGM of any period q is guaranteedby Theorem 4.If π d − d +2 ≤ a ≤ π d +1 d − then − d ≤ ˆ Q ISq ,a ( π )ˆ Q ISq ,a (0) ≤ d . This means that | ˆ Q ISq ,a ( k )ˆ Q ISq ,a (0) | > d is only possible if ˆ Q ISq ,a ( k )ˆ Q ISq ,a (0) > d . Solving this inequality for k ∈ [0 , π ] gives the condition k < π − (cid:114) π d ) − a (1 − d ) . (49)Substituting k = πq , we obtain the minimal period q ( a, d ) for the Inversesquare-model in that case.Finally, if a > π d +1 d − then ˆ Q ISq ,a ( π )ˆ Q ISq ,a (0) < − d . In this case, existence of 2-height periodic n-t.i. GGM is guaranteed and also those (positive) partsof the spectrum of D S q satisfying (49) will correspond to height periodicnon-t.i. GGM. Moreover, ˆ Q ISq ,a (2 π/ Q ISq ,a (0) < − d if and only if a > π d +1 d − .In this case, also q -height periodic n-t.i. GGM for q = 3 and hence, bymonotonicity, for all q ≥ References [1] M. Aizenman, “Translation invariance and instability of phase coexistence in thetwo-dimensional Ising system,”
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