Gradient Gibbs measures for the SOS model

# Gradient Gibbs measures for the SOS model with countable values on a Cayley tree

F. Henning, C. Külske, A. Le Ny, U. A. Rozikov F. Henning
Fakultät für Mathematik, Ruhr-University of Bochum, Postfach 102148,  44721, Bochum, Germany.
C. Külske
Fakultät für Mathematik, Ruhr-University of Bochum, Postfach 102148,  44721, Bochum, Germany.
A.  Le Ny
Université Paris-Est, Laboratoire d’Analyse et de Mathématiques Appliquées, LAMA UMR CNRS 8050, UPEC, 91 Avenue du Général de Gaulle, 94010 Créteil cedex, France.
U. A. Rozikov
Institute of mathematics, 81, Mirzo Ulug’bek str., 100125, Tashkent, Uzbekistan.
###### Abstract.

We consider an SOS (solid-on-solid) model, with spin values from the set of all integers, on a Cayley tree of order and are interested in translation-invariant gradient Gibbs measures (GGMs) of the model. Such a measure corresponds to a boundary law (a function defined on vertices of the Cayley tree) satisfying a functional equation. In the ferromagnetic SOS case on the binary tree we find up to five solutions to a class of -periodic boundary law equations (in particular, some two periodic ones). We show that these boundary laws define up to four distinct GGMs. Moreover, we construct some -periodic boundary laws on the Cayley tree of arbitrary order , which define GGMs different from the -periodic ones.

Mathematics Subject Classifications (2010). 82B26 (primary); 60K35 (secondary)

Key words. SOS model, Cayley tree, Gibbs measure, tree-indexed Markov chain, gradient Gibbs measures, boundary law.

## 1. Introduction

We consider models where an infinite-volume spin-configuration is a function from the vertices of the tree Cayley to the local configuration space .

A solid-on-solid (SOS) model is a spin system with spins taking values in (a subset of) the integers, and formal Hamiltonian

 H(σ)=−J∑⟨x,y⟩|ω(x)−ω(y)|,

where is a coupling constant. As usual, denotes a pair of nearest neighbour vertices.

For the local configuration space we consider in the present paper the full set . The model can be considered as a generalization of the Ising model, which corresponds to , or a less symmetric variant of the Potts model with non-compact state space. SOS-models on the cubic lattice were analyzed in [14] where an analogue of the so-called Dinaburg–Mazel–Sinai theory was developed. Besides interesting phase transitions in these models, the attention to them is motivated by applications, in particular in the theory of communication networks; see, e.g., [11], [16]. SOS models with have also been used as simplified discrete interface models which should approximate the behaviour of a Dobrushin-state in an Ising model when the underlying graph is , and . There is the issue of possible non-existence of any Gibbs measure in the case of such unbounded spins, in particular in the additional presence of disorder (see [3] and [4]). In this paper we show that on the Cayley tree there are several translation invariant gradient Gibbs measures. For more background on Gradient Gibbs measures on the lattice, also in the case of real valued state space, we refer to [9], [1], [8], [5], [6] and [2].

Compared to the Potts model, the -state SOS model has less symmetry: The full symmetry of the Hamiltonian under joint permutation of the spin values is reduced to the mirror symmetry, which is the invariance of the model under the map on the local spin space. Therefore one expects a more diverse structure of phases.

To the best of our knowledge, the first paper devoted to the SOS model on the Cayley tree is [17]. In [17] the case of arbitrary is treated and a vector-valued functional equation for possible boundary laws of the model is obtained. Recall that each solution to this functional equation determines a splitting Gibbs measure (SGM), in other words a tree-indexed Markov chain which is also a Gibbs measure. Such measures can be obtained by propagating spin values along the edges of the tree, from any site singled out to be the root to the outside, with a transition matrix depending on initial Hamiltonian and the boundary law solution. In particular the homogeneous (site-independent) boundary laws then define translation-invariant (TI) SGMs. For a recent investigation of the influence of weakly non-local perturbations in the interaction to the structure of Gibbs measures, see [2] in the context of the Ising model.

Also the symmetry (or absence of symmetry) of the Gibbs measures under spin reflection is seen in terms of the corresponding boundary law. For SOS models some TISGMs which are symmetric have already been studied in the particular case in [17], and in [18]. In [12], for , a detailed description of TISGMs (symmetric and non-symmetric ones) is given: it is shown the uniqueness in the case of antiferromagnetic interactions, and existence of up to seven TISGMs in the case of ferromagnetic interactions. See also [19] for more details about SOS models on trees.

In the situation of an unbounded local spin space the normalisability condition given in [21] (which is needed to construct a SGM, in other words a tree indexed Markov chain, from a given boundary law solution) is not automatically satisfied anymore. In this paper we are interested in the class of (spatially homogeneous/ tree-automorphism invariant) height-periodic boundary laws to tree-automorphism invariant potentials whose elements violate this normalisability condition. Here, a spatially homogeneous height-periodic boundary law with period is a -periodic function on the local state space . Although the procedure of constructing a Gibbs measures from boundary laws described in [21] can not be applied to elements of that class, we are still able to assign a translational invariant gradient Gibbs measure (GGM) on the space of gradient configurations to each such spatially homogeneous height-periodic boundary law, compare [13]. This motivates the study of spatially homogeneous height-periodic boundary laws as useful finite-dimensional objects which are are easier to handle than the non-periodic ones required to fulfill the normalisability condition. Gradient Gibbs measures describe height differences, Gibbs measures describe absolute heights. Each Gibbs measure defines a gradient Gibbs measures, but the converse in not true, which is a phenomenon that is well-known from the lattice. Some more explanation will be given in the following sections. The main goal of this paper then consists in the description of a class of boundary solutions which have periods of , and with respect to shift in the height direction on the local state space , and their associated GGMs.

The paper is organized as follows. In Section 2 we first present the preliminaries of our model. Section 3 then contains a summary on the notion of GGMs on trees and their construction from homogeneous periodic boundary laws. For further details see [13]. The main part, section 4, is devoted to the description of a set of homogeneous , and -periodic boundary laws. Solving the associated boundary law equations for the -periodic and the -periodic case on the binary tree we prove that depending on the system parameters this set contains one up to five elements, yet the number of distinct GGMs assigned to them will turn out to be at most four. In the last subsection we construct GGMs for 3-periodic boundary laws on the -regular tree for arbitrary .

## 2. Preliminaries

Cayley tree. The Cayley tree of order (or -regular tree) is an infinite tree, i.e. a locally finite connected graph without cycles, such that exactly edges originate from each vertex. Let where is the set of vertices and the set of edges. Two vertices are called nearest neighbours if there exists an edge connecting them. We will use the notation . A collection of nearest neighbour pairs is called a path from to . The distance on the Cayley tree is the number of edges of the shortest path from to .

Furthermore, for any we define its outer boundary as

 ∂Λ:={x∉Λ:d(x,y)=1 for some y∈Λ}.

SOS model. We consider a model where the spin takes values in the set of all integer numbers , and is assigned to the vertices of the tree. A (height) configuration on is then defined as a function ; the set of all height configurations is . Take the power set as measurable structure on and then endow with the product -algebra where denotes the projection on the th coordinate. We also sometimes consider more general finite subsets of the tree and we write for the set of all those finite subtrees.
Recall here that the (formal) Hamiltonian of the SOS model is

 H(σ)=−J∑⟨x,y⟩∈L|ωx−ωy|, (2.1)

where is a constant which we will set to (incorporated in the inverse temperature ) in the following. As defined above, denotes nearest neighbour vertices.

Note that the above Hamiltonian depends only on the height difference between neighbouring vertices but not on absolute heights (it is given by a gradient interaction potential in the terminology of [13]). This suggests reducing complexity of the configuration space by considering gradient configurations instead of height configurations as it will be explained in the following section.

## 3. Gradient Gibbs measures and an infinite system of functional equations

Gradient configurations: Let the Cayley tree be called . We may induce an orientation on relative to an arbitrary site (which we may call the root) by calling an edge oriented iff it points away from the . More precisely, the set of oriented edges is defined by

 →L:=→Lρ:={⟨x,y⟩∈L:d(ρ,y)=d(ρ,x)+1}.

Note that the oriented graph also possesses all tree-properties, namely connectedness and absence of loops.
For any height configuration and the height difference along the edge is given by and we also call the gradient field of . The gradient spin variables are now defined by for each . Let us denote the space of gradient configurations by . Equip the integers with the power set as measurable structure. Having done this, the measurable structure on the space is given by the product -algebra . Clearly then becomes a measurable map.

For any fixed site and given spin value , each gradient configuration (uniquely) determines a height configuration by the measurable map

 φx,ωx:{Ω∇→Ω(φx,ωx(ζ))y=ωx+∑b∈Γ(x,y)ζb, (3.1)

where is the unique path from to . From this we get the following two statements:

1. The linear map is surjective and

2. The kernel of is given by the spatially homogeneous configurations.

Therefore we have the identification

 Ω∇=Z→L=ZV/Z. (3.2)

Here, is meant in the sense of isomorphy between Abelian groups. Endowing with the final -algebra generated by the respective coset projection we can also regard this isomorphy as an isomorphy between measurable spaces due to measurability of the maps and .

Note that statement (1) above relies on the absence of loops in trees. For gradient configurations on lattices in more than one dimension a further plaquette condition is needed (see [9]). In contrast to this, its following statement (2) is based on connectedness of the tree. Therefore for any finite subtree the isomorphy (3.2) between measurable spaces restricts to an isomorphy between and and , where the sets are endowed with the respective final and product -algebra.

Further note that for any the bijection

 {ZV→Z→L×Zω=(ωx)x∈V↦(∇ω,ωw) (3.3)

is an isomorphism with respect to the product -algebra on , where the inverse map is given by (3.1). In the following, this will allow us to easily identify any measure on with its push forward on the space .

Gibbs measure: Recall that the set of height configurations was endowed with the product -algebra , where denotes the power set of . Then for any consider the coordinate projection map and the -algebra of cylinder sets on generated by the map .

Now we are ready to define Gibbs measures on the space of height-configurations for the model (2.1) on a Cayley tree. Let be a -finite positive fixed a-priori measure, which in the following we will always assume to be the counting measure.

Gibbs measures are built within the DLR framework by describing conditional probabilities w.r.t. the outside of finite sets, where a boundary condition is frozen. One introduces a so-called Gibbsian specification so that any Gibbs measure specified by verifies

 μ(A|FΛc)=γΛ(A|⋅)μ−a.s. (3.4)

for all and . The Gibbsian specification associated to a potential is given at any inverse temperature , for any boundary condition as

 γΛ(A|ω)=1Zβ,ΦΛ∫e−βHΦΛ(σΛωΛc)1A(σΛωΛc)ν⊗Λ(dσΛ), (3.5)

where the partition function – that has to be non-null and convergent in this countable infinite state-space context (this means that is -admissible in the terminology of [10])– is the standard normalization whose logarithm is often related to pressure or free energy.

In our SOS-model on the Cayley tree is the unbounded nearest neighbour potential with
and , so is a Markov specification in the sense that

 (3.6)

In order to build up gradient specifications from the Gibbsian specifications defined above, we need to consider the following: Due to the absence of loops in trees, for any finite the complement is not connected but consists of at least two connected components where each of these contains at least one element of . This means that the gradient field outside does not contain any information on the relative height of the boundary (which is to be understood as an element of ). More precisely, let denote the number of connected components in and note that .

Applying (3.1) to each connected component, an analogue to (3.2) becomes

 Z{b∈→L|b⊂Λc}×(Z∂Λ/Z)⊃Z{b∈→L|b⊂Λc}×(Zcc(Λc)/Z)=ZΛc/Z⊂ZV/Z. (3.7)

where ”” is in the sense of isomorphy between measurable spaces. For any let denote the image of under the coordinate projection with the latter set endowed with the final -algebra generated by the coset projection. Set

 F∇Λ:=σ((ηb)b⊂Λc)⊂T∇Λ:=σ((ηb)b⊂Λc,[η]∂Λ). (3.8)

Then contains all information on the gradient spin variables outside and also information on the relative height of the boundary . By (3.7) we have that for any event the -measurable function is also measurable with respect to , but in general not with respect to . These observations lead to the following:

###### Definition 1.

The gradient Gibbs specification is defined as the family of probability kernels from to such that

 ∫F(ρ)γ′Λ(dρ∣ζ)=∫F(∇φ)γΛ(dφ∣ω) (3.9)

for all bounded -measurable functions , where is any height-configuration with .

Using the sigma-algebra , this is now a proper and consistent family of probability kernels, i.e.

 γ′Λ(A∣ζ)=1A(ζ) (3.10)

for every and for any finite volumes with . The proof is similar to the situation of regular (local) Gibbs specifications [10, Proposition 2.5].

Let be the set of bounded functions on . Gradient Gibbs measures will now be defined in the usual way by having their conditional probabilities outside finite regions prescribed by the gradient Gibbs specification:

###### Definition 2.

A measure is called a gradient Gibbs measure (GGM) if it satisfies the DLR equation

 ∫ν(dζ)F(ζ)=∫ν(dζ)∫γ′Λ(d~ζ∣ζ)F(~ζ) (3.11)

for every finite and for all . The set of gradient Gibbs measures will be denoted by .

Construction of GGMs via boundary laws:

In what follows we may assume the a-priori measure on to be the counting measure. On trees with nearest-neighbours potentials such as the one we consider here, it is possible to use the natural orientations of edges to introduce tree-indexed Markov chains. These are probability measures having the property that for -a.e. oriented edges and any ,

 μ(σy=ωy|F(−∞,xy))=μ(σy=ωy|Fx),

where

 (−∞,xy):={w∈V∣⟨x,y⟩∈→Lw},

denotes the past of the edge . One can associate to a transition matrix defined to be any stochastic matrix satisfying for all

 μ(σy=ωy|Fx)=Pxy(σx,ωy)μ−a.s.

For interaction potential , where bonds are denoted , one first defines symmetric transfer matrices following the terminology of Cox [7] or Zachary [21, 22] (see also [10]). Setting

 Qb(ωb)=e−(Φb(ωb)+|∂x|−1Φ{x}(ωx)+|∂y|−1Φ{y}(ωy))

one can rewrite the Gibbsian specification as

 γΦΛ(σΛ=ωΛ|ω)=(ZΦΛ)(ω)−1∏b∩Λ≠∅Qb(ωb).

If for any bond the transfer operator is a function of gradient spin variable we call the underlying potential a gradient interaction potential.
Now we note the following: On the one hand, each extreme Gibbs measure on a tree with respect to a Markov specification is a tree-indexed Markov chain (Theorem 12.6 in [10]). On the other hand (Lemma 3.1 in[21]), a measure is a Gibbs measure with respect to a nearest neighbour potential with associated family of transfer matrices iff its marginals at any finite volume are of the form

 μ(σΛ∪∂Λ=ωΛ∪∂Λ)=cΛ(ω∂Λ)∏b∩Λ≠∅Qb(ωb) (3.12)

for some function . Taking this into account leads to the concept of boundary laws that allows to describe the Gibbs measures that are Markov chains on trees.

###### Definition 3.

A family of vectors with is called a boundary law for the transfer operators if for each there exists a constant such that the consistency equation

 lxy(ωx)=cxy∏z∈∂x∖{y}∑ωz∈ZQzx(ωx,ωz)lzx(ωz) (3.13)

holds for every . A boundary law is called to be -periodic if for every oriented edge and each .

In our unbounded discrete context, there is as in the finite-state space context, a one-to-one correspondence between boundary laws and tree-indexed Markov chains, but for some boundary laws only, the ones that are normalisable in the sense of Zachary [21, 22].

###### Definition 4 (Normalisable boundary laws).

A boundary law is said to be normalisable if and only if

 ∑ωx∈Z(∏z∈∂x∑ωz∈ZQzx(ωx,ωz)lzx(ωz))<∞ (3.14)

at any .

The correspondence now reads the following:

###### Theorem 1 (Theorem 3.2 in [21]).

For any Markov specification with associated family of transfer matrices we have

1. Each normalisable boundary law for defines a unique tree-indexed Markov chain via the equation given for any connected set

 μ(σΛ∪∂Λ=ωΛ∪∂Λ)=(ZΛ)−1∏y∈∂ΛlyyΛ(ωy)∏b∩Λ≠∅Qb(ωb), (3.15)

where for any , denotes the unique of in .

2. Conversely, every tree-indexed Markov chains admits a representation of the form (3.15) in terms of a normalisable boundary law (unique up to a constant positive factor).

###### Remark 1.

The Markov chain defined in (3.15) has the transition probabilities

 Pij(ωi,x)=μ(σj=x∣σi=ωi)=lji(x)Qji(x,ωi)∑ylji(y)Qji(y,ωi). (3.16)

The expressions (3.16) may exist even in situations where the underlying boundary law is not normalisable in the sense of Definition 4. However, the Markov chain given by the so defined transition probabilities is in general not positively recurrent which means that it does not possess an invariant probability measure. More precisely if the Markov chain defined by (3.16) is of the form (3.15) (and hence of the form (3.12)) then its underlying boundary law must be necessarily normalisable as one can see by considering (3.15) for . Thus, there is no obvious extension of Theorem 1 to non-normalisable boundary laws.

Let us now assume that for all (this holds obviously true for the SOS model). We call a vector a (spatially homogeneous) boundary law if there exists a constant such that the consistency equation

 l(i)=c(∑j∈ZQ(i,j)l(j))k (3.17)

is satisfied for every .
Now assume that the elements of the family do not depend on the bonds i.e. for all , i.e. the underlying potential is tree-automorphism invariant.

In the case of spatially homogeneous boundary laws the expression (3.14) in the definition of normalisability reads

 ∑i∈Z(∑j∈ZQ(i,j)l(j))k+1=∑i∈Zc−k+1k(c(∑j∈ZQ(i,j)l(j))k)k+1k=c−k+1k∑i∈Z(l(i))k+1k,

which means that any spatially homogeneous normalisable boundary law is an element of the space . Thus periodic spatially homogeneous boundary laws are never normalisable in the sense of Definition 4.
However, it is possible to assign (tree-automorphism invariant) Gradient Gibbs measures to spatially homogeneous -periodic boundary laws to tree-automorphism invariant gradient interaction potentials. The main idea consists in considering for any boundary law to a gradient interaction potential and any finite connected subset the (in general only -finite) measure on given by the assignment (3.15), i.e.

 μ(σΛ∪∂Λ=ωΛ∪∂Λ)=∏y∈∂ΛlyyΛ(ωy)∏b∩Λ≠∅Qb(ωb). (3.18)

Then fix any pinning site and identify with its pushforward measure on under (3.3). This measure has the marginals

 μΛ∪∂Λ(σw=i,ηΛ∪∂Λ=ζΛ∪∂Λ)=μΛ∪∂Λ(σw=i)μΛ∪∂Λ(ηΛ∪∂Λ=ζΛ∪∂Λ∣σw=i)=μΛ∪∂Λ(σw=i)∏y∈∂ΛlyyΛ(i+∑b∈Γ(w,y)ζb)∏b∩Λ≠∅Qb(ζb). (3.19)

If the boundary law is assumed to be -periodic, then will depend on only modulo . For any class label this allows us to obtain a probability measure on by setting

 νw,s(ηΛ∪∂Λ=ζΛ∪∂Λ):=ZΛw,sμΛ∪∂Λ(ηΛ∪∂Λ=ζΛ∪∂Λ∣σw=s)=ZΛw,s∏y∈∂ΛlyyΛ(Tq(s+∑b∈Γ(w,y)ζb))∏b∩Λ≠∅Qb(ζb), (3.20)

where is a normalization constant and denotes the coset projection. Then one can show the following:

###### Theorem 2 (Theorem 3.1 in [13]).

Let be any -periodic boundary law to some gradient interaction potential. Fix any site and any class label . Then the definition

 νw,s(ηΛ∪∂Λ=ζΛ∪∂Λ)=ZΛw,s∏y∈∂ΛlyyΛ(Tq(s+∑b∈Γ(w,y)ζb))∏b∩Λ≠∅Qb(ζb), (3.21)

where with is any finite connected set, and is a normalization constant, gives a consistent family of probability measures on the gradient space . The measures will be called pinned gradient measures.

By construction, the pinned gradient measures on have a restricted gradient (Gibbs) property in the sense that the DLR-equation (3.11) holds for any finite which does not contain the pinning site (for details see [13]). If the -periodic boundary law is now additionally spatially homogeneous and the underlying potential is tree-automorphism invariant then it is possible to obtain a tree-automorphism invariant probability measure on the the gradient space by mixing the pinned gradient measures over an appropriate distribution on . In this case, the restricted gradient Gibbs property of each of the pinned gradient measures leads to the Gibbs property of the measure .

A useful representation of the finite-volume marginals of the resulting GGM is given in the following theorem:

###### Theorem 3 (Theorem 4.1, Remark 4.2 in [13]).

Let be any spatially homogeneous -periodic boundary law to a tree-automorphism invariant gradient interaction potential on the Cayley tree. Let be any finite connected set and let be any vertex. Then the measure with marginals given by

 ν(ηΛ∪∂Λ=ζΛ∪∂Λ)=ZΛ ⎛⎝∑s∈Zq∏y∈∂Λl(s+∑b∈Γ(w,y)ζb)⎞⎠∏b∩Λ≠∅Q(ζb), (3.22)

where is a normalisation constant, defines a spatially homogeneous GGM on .

###### Remark 2.

Setting the marginals of the measure defined in Theorem 3 can be written in the form:

 ν(ηΛ∪∂Λ=ζΛ∪∂Λ)=ZΛ(∑j∈Zq∏i∈Zqlnwi+j(ζΛ∪∂Λ)i)∏b∩Λ≠∅Q(ζb)=ZΛ(∑j∈Zq∏i∈Zqlnwi(ζΛ∪∂Λ)i+j)∏b∩Λ≠∅Q(ζb). (3.23)

This representation directly shows that two periodic boundary laws will lead to the same GGM if one is obtained from the other by a cyclic permutation or multiplication with a positive constant.
To obtain sufficient criteria for two GGM and associated to two distinct periodic boundary laws and with being distinct we first observe that if and only if

 ∑j∈Zq∏i∈Zqlnwi+j(ζΛ∪∂Λ)i∑j∈Zq∏i∈Zq~lnwi+j(ζΛ∪∂Λ)i=Z~lΛZlΛ (3.24)

for all finite subtrees and .
Thus if and only if for any finite subtree there is a constant such that

 ∑j∈Zq∏i∈Zqlni+ji=c(Λ)∑j∈Zq∏i∈Zq~lni+ji (3.25)

for all vectors with . If we take a single-bond volume , where , we obtain the marginal

 ν(ηb=ζb)=Zb∑s∈Zql(s)l(s+ζb)Q(ζb)=Zb(∑j∈Zq∏i∈Zqlni+j(ζb)i)Qb(ζb). (3.26)

From this we get that if then condition (3.25) is fulfilled for all vectors
with .

We will now conclude some statements on identifiability of GGM with respect to the class of boundary laws which we will describe in the following section.

###### Lemma 1.

Let and be two -periodic boundary laws with . Denote , .
Then

 νl=ν~lif and only ifa1=a2 or a1a2=1 (3.27)
###### Proof.

Let us first prove that if . Using the marginals representation given in Remark 2 we have that if and only if

 ∑j∈{0,1}∏i∈{0,1}lnwi+j(ζΛ∪∂Λ)i∑j∈{0,1}∏i∈{0,1}~lnwi+j(ζΛ∪∂Λ)i=const(Λ∪∂Λ) (3.28)

for any finite subtree and . Let then the vectors are of the form for some integer . Inserting this into (3.28) we conclude that if and only if for all which can be realized as the number of points in the boundary of a finite subtree and any we have:

 an−m11+am11an−m12+am12=an−m21+am11an−m22+am22. (3.29)

We may further assume and write . Then this equation reduces to

 an1+1an2+1=an−m1+am1an−m2+am2, (3.30)

which holds true if .
To prove the other direction we must show that and are the only solutions to the system (3.30):
For any set . Then (3.30) is equivalent to . Consider any . Clearly is continuous, strictly decreasing on and strictly increasing on , which means that for any there is at most one with . Since we have that if and only if or .

###### Lemma 2.

Consider any -periodic boundary law of the type

 l(a,b)i=⎧⎪⎨⎪⎩1,  if  i≡0  %or  2mod4a,  if  i≡1mod4b,  if  i≡3mod4

and denote the associated GGM by . Let be two such boundary laws. If then necessarily

 a1+b1=a2+b2or (ai+bi)(aj+bj)=4.
###### Proof.

Consider the marginal on a set , where is any edge. Inserting the vectors and into (3.25) we conclude that if then there is some constant with

1. ,

2. .

Adding twice the third equation to the first we obtain

 (a1+b1)2+4=c((a2+b2)2+4),

which in combination with gives

 (a1+b1)2+4(a2+b2)2+4=a1+b1a2+b2. (3.31)

Setting and leads to the equation which is equivalent to

 (x−y)(xy−4)=0.

This completes the proof. ∎

###### Lemma 3.

Consider the -regular tree, , and a -periodic boundary law of the type

 l(c)i={1,  if  i≡0mod3c,  else.

Denote the associated GGM by . Then the following holds true:

1. [a)]

2. if and only if for any we have for all with , where

 f(m0,m1,m2)(x):=xm0+xm1+xm21+2xn(k−1)+2,x>0.
3. if and only if .

4. The GGMs associated to the nontrivial members of this family of solutions are all different from the GGMs associated to the solutions given by the family of boundary laws defined in Lemma 2.

###### Proof.

The structure of the proof is similar to the proof of Lemma 2:

1. [a)]

2. First note that for any subtree of the -regular tree with vertices we have points in the outer boundary which follows by induction on (see [20]). Thus if and only if for each the equation (3.25) holds true for all with . This is equivalent to the existence of some depending only on and with

 c(n0+n1)1+c(n0+n2)1+c(n1+n2)1=λ(c(n0+n1)2+c(n0+n2)2+c(n1+n2)2).

for all such vectors .
Setting , i.e. this is equivalent to

 cm01+cm11+cm21=λ(cm02+c