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

We consider an SOS (solid-on-solid) model, with spin values from the set of all integers, on a Cayley tree of order k 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 4-periodic boundary law equations (in particular, some 2-periodic ones). We show that these boundary laws define up to four distinct GGMs. Moreover, we construct some 3-periodic boundary laws on the Cayley tree of arbitrary order k, which define GGMs different from the 4-periodic ones.


Introduction
We consider models where an infinite-volume spin-configuration ω is a function from the vertices of the tree Cayley to the local configuration space E ⊆ Z.
A solid-on-solid (SOS) model is a spin system with spins taking values in (a subset of) the integers, and formal Hamiltonian where J ∈ R is a coupling constant. As usual, x, y denotes a pair of nearest neighbour vertices.
For the local configuration space we consider in the present paper the full set E := Z. The model can be considered as a generalization of the Ising model, which corresponds to E = {−1, 1}, 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 socalled 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 E = Z 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 Z d , and d ≥ 2. 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 m-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 ω i → m − ω i 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 m ≥ 1 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 m = 2 in [17], and m = 3 in [18]. In [12], for m = 2, 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 heightperiodic boundary law with period q is a q-periodic function on the local state space Z. 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 2, 3 and 4 with respect to shift in the height direction on the local state space Z, 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 2, 3 and 4-periodic boundary laws. Solving the associated boundary law equations for the 2-periodic and the 4-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 k-regular tree for arbitrary k ≥ 2.

Preliminaries
Cayley tree. The Cayley tree Γ k of order k ≥ 1 (or k-regular tree) is an infinite tree, i.e. a locally finite connected graph without cycles, such that exactly k + 1 edges originate from each vertex. Let Γ k = (V, L) where V is the set of vertices and L the set of edges. Two vertices x, y ∈ V are called nearest neighbours if there exists an edge l ∈ L connecting them. We will use the notation l = x, y . A collection of nearest neighbour pairs x, x 1 , x 1 , x 2 , ..., x d−1 , y is called a path from x to y. The distance d(x, y) on the Cayley tree is the number of edges of the shortest path from x to y.
SOS model. We consider a model where the spin takes values in the set of all integer numbers Z := {. . . , −1, 0, 1, . . . }, and is assigned to the vertices of the tree. A (height) configuration ω on V is then defined as a function x ∈ V → ω x ∈ Z; the set of all height configurations is Ω := Z V . Take the power set 2 Z as measurable structure on Z and then endow Ω with the product σ-algebra F := σ{ω i | i ∈ V } where ω i : Ω → Z denotes the projection on the ith coordinate. We also sometimes consider more general finite subsets Λ of the tree and we write S for the set of all those finite subtrees. Recall here that the (formal) Hamiltonian of the SOS model is where J ∈ R is a constant which we will set to 1 (incorporated in the inverse temperature β) in the following. As defined above, x, y 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.

Gradient Gibbs measures and an infinite system of functional equations
Gradient configurations: Let the Cayley tree be called Γ k . We may induce an orientation on Γ k relative to an arbitrary site ρ (which we may call the root) by calling an edge x, y oriented iff it points away from the ρ. More precisely, the set of oriented edges is defined by Note that the oriented graph (V, L) also possesses all tree-properties, namely connectedness and absence of loops. For any height configuration ω = (ω(x)) x∈V ∈ Z V and b = x, y ∈ L the height difference along the edge b is given by ∇ω b = ω y − ω x and we also call ∇ω the gradient field of ω. The gradient spin variables are now defined by η x,y = ω y − ω x for each x, y ∈ L.
Let us denote the space of gradient configurations by Ω ∇ = Z L . Equip the integers Z with the power set as measurable structure. Having done this, the measurable structure on the space Ω ∇ is given by the product σ-algebra F ∇ := σ({η b | b ∈ L}). Clearly ∇ : (Ω, F) → (Ω ∇ , F ∇ ) then becomes a measurable map.
For any fixed site x ∈ V and given spin value ω x ∈ Z, each gradient configuration ζ ∈ Ω ∇ (uniquely) determines a height configuration by the measurable map where Γ(x, y) is the unique path from x to y. From this we get the following two statements: (1) The linear map ∇ : Z V → Z L is surjective and (2) The kernel of ∇ is given by the spatially homogeneous configurations.
Therefore we have the identification Here, = is meant in the sense of isomorphy between Abelian groups. Endowing Z V /Z with the final σ-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 ϕ x,ωx 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 Λ ∈ S the isomorphy (3.2) between measurable spaces restricts to an isomorphy between Z Λ /Z and Z {b∈ L|b⊂Λ} and Z Λ /Z, where the sets are endowed with the respective final and product σ-algebra.
Further note that for any w ∈ V the bijection is an isomorphism with respect to the product σ-algebra on Z L × Z, where the inverse map is given by (3.1). In the following, this will allow us to easily identify any measure on Z V with its push forward on the space Z L × Z.
Gibbs measure: Recall that the set of height configurations Ω := Z V was endowed with the product σ-algebra ⊗ i∈V 2 Z , where 2 Z denotes the power set of Z. Then for any Λ ⊂ V consider the coordinate projection map σ Λ : Z V → Z Λ and the σ-algebra F Λ := σ(ω Λ ) of cylinder sets on Z V generated by the map ω Λ . Now we are ready to define Gibbs measures on the space of height-configurations for the model (2.1) on a Cayley tree. Let ν = {ν(i) > 0, i ∈ Z} be a σ-finite positive fixed apriori 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 γ so that any Gibbs measure µ ∈ G(γ) specified by γ verifies for all Λ ∈ S and A ∈ F. The Gibbsian specification associated to a potential Φ is given at any inverse temperature β > 0, for any boundary condition ω ∈ Ω as where the partition function Z β,Φ Λ -that has to be non-null and convergent in this countable infinite state-space context (this means that Φ is ν-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 Φ is the unbounded nearest neighbour potential with 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 Λ ⊂ Z the complement Λ c is not connected but consists of at least two connected components where each of these contains at least one element of ∂Λ. This means that the gradient field outside Λ does not contain any information on the relative height of the boundary ∂Λ (which is to be understood as an element of Z ∂Λ /Z). More precisely, let cc(Λ c ) denote the number of connected components in Λ c and note that 2 ≤ cc(Λ c ) ≤ |∂Λ|.
Applying (3.1) to each connected component, an analogue to (3.2) becomes where "=" is in the sense of isomorphy between measurable spaces. For any η ∈ Ω ∇ = Z V /Z let ⌊η⌋ ∂Λ ∈ Z ∂Λ /Z denote the image of η under the coordinate projection Z V /Z → Z ∂Λ /Z with the latter set endowed with the final σ-algebra generated by the coset projection. Set Then T ∇ Λ contains all information on the gradient spin variables outside Λ and also information on the relative height of the boundary ∂Λ. By (3.7) we have that for any event A ∈ F ∇ the F Λ c -measurable function γ Λ (A|·) is also measurable with respect to T ∇ Λ , but in general not with respect to F ∇ Λ . These observations lead to the following: Definition 1. The gradient Gibbs specification is defined as the family of probability Using the sigma-algebra T ∇ Λ , this is now a proper and consistent family of probability kernels, i.e.
The proof is similar to the situation of regular (local) Gibbs specifications [10, Proposition 2.5].
Let C b (Ω ∇ ) be the set of bounded functions on Ω ∇ . 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: for every finite Λ ⊂ V and for all F ∈ C b (Ω ∇ ). The set of gradient Gibbs measures will be denoted by G ∇ (γ).

Construction of GGMs via boundary laws:
In what follows we may assume the a-priori measure ν on Z to be the counting measure. On trees with nearest-neighbours potentials Φ 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 µ having the property that for µ-a.e. oriented edges xy and any ω y ∈ E, denotes the past of the edge x, y . One can associate to µ a transition matrix defined to be any stochastic matrix P = (P xy ) xy satisfying for all ω y ∈ E where bonds are denoted b = x, y , one first defines symmetric transfer matrices Q b following the terminology of Cox [7] or Zachary [21,22] (see also [10]). Setting one can rewrite the Gibbsian specification as If for any bond b = x, y the transfer operator Q b (ω b ) is a function of gradient spin variable ζ b = ω y − ω x we call the underlying potential Φ 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 µ is a Gibbs measure with respect to a nearest neighbour potential Φ with associated family of transfer matrices (Q b ) b∈L iff its marginals at any finite volume Λ ⊂ V are of the form for some function c Λ : ∂Λ → R + . 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 {l xy } x,y ∈ E with l xy ∈ (0, ∞) Z is called a boundary law for the transfer operators {Q b } b∈L if for each x, y ∈ L there exists a constant c xy > 0 such that the consistency equation holds for every ω x ∈ Z. A boundary law is called to be q-periodic if l xy (ω x + q) = l xy (ω x ) for every oriented edge x, y ∈ L and each ω x ∈ Z.
In our unbounded discrete context, there is as in the finite-state space context, a oneto-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 l is said to be normalisable if and only if at any x ∈ V .
The correspondence now reads the following: Theorem 1 (Theorem 3.2 in [21]). For any Markov specification γ with associated family of transfer matrices (Q b ) b∈L we have (1) Each normalisable boundary law (l xy ) x,y for (Q b ) b∈L defines a unique treeindexed Markov chain µ ∈ G(γ) via the equation given for any connected set where for any y ∈ ∂Λ, y Λ denotes the unique n.n. of y in Λ.
. Let us now assume that Q b = Q for all b ∈ L (this holds obviously true for the SOS model). We call a vector l ∈ (0, ∞) Z a (spatially homogeneous) boundary law if there exists a constant c > 0 such that the consistency equation is satisfied for every i ∈ Z. Now assume that the elements of the family (Q b ) b∈L do not depend on the bonds i.e. Q b = Q for all b ∈ L, 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∈Z which means that any spatially homogeneous normalisable boundary law is an element of the space l 1+ 1 k . 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 q-periodic boundary laws to tree-automorphism invariant gradient interaction potentials. The main idea consists in considering for any boundary law (l xy ) to a gradient interaction potential and any finite connected subset Λ ⊂ V the (in general only σ-finite) measure µ Λ on (Z Λ∪∂Λ , ⊗ i∈Λ∪∂Λ 2 Z ) given by the assignment (3.15) Then fix any pinning site w ∈ Λ and identify µ Λ with its pushforward measure on Z L × Z under (3.3). This measure has the marginals If the boundary law l is assumed to be q-periodic, then µ Λ∪∂Λ (η Λ∪∂Λ = · | σ w = i) will depend on i only modulo q. For any class label s ∈ Z q this allows us to obtain a probability measure ν w,s on Z {b∈ L|b⊂Λ} by setting where Z Λ w,s is a normalization constant and T q : Z → Z q denotes the coset projection. Then one can show the following: Theorem 2 (Theorem 3.1 in [13]). Let (l <xy> ) <x,y>∈ L be any q-periodic boundary law to some gradient interaction potential. Fix any site w ∈ V and any class label s ∈ Z q .
Then the definition where Λ with w ∈ Λ ⊂ V is any finite connected set, ζ Λ∪∂Λ ∈ Z {b∈ L|b⊂(Λ∪∂Λ)} and Z Λ w,s is a normalization constant, gives a consistent family of probability measures on the gradient space Ω ∇ . The measures ν w,s will be called pinned gradient measures.
By construction, the pinned gradient measures ν w,s on Ω ∇ have a restricted gradient (Gibbs) property in the sense that the DLR-equation (3.11) holds for any finite Λ ⊂ V which does not contain the pinning site w (for details see [13]). If the q-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 ν on the the gradient space by mixing the pinned gradient measures over an appropriate distribution on Z q . In this case, the restricted gradient Gibbs property of each of the pinned gradient measures leads to the Gibbs property of the measure ν.
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 l be any spatially homogeneous q-periodic boundary law to a tree-automorphism invariant gradient interaction potential on the Cayley tree. Let Λ ⊂ V be any finite connected set and let w ∈ Λ be any vertex. Then the measure ν with marginals given by

22)
where Z Λ is a normalisation constant, defines a spatially homogeneous GGM on Ω ∇ .

(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 ν l and νl associated to two distinct periodic boundary laws l andl with l 0 =l 0 = 1 being distinct we first observe that ν l = νl if and for all finite subtrees Λ and ζ Λ∪∂Λ ∈ Z {b∈L|b⊂Λ∩∂Λ} . Thus ν l = νl if and only if for any finite subtree Λ there is a constant c(Λ) > 0 such that j∈Zq i∈Zq for all vectors (n 0 , n 2 , . . . , n q−1 ) ∈ N 0 with i∈Zq n i = |∂Λ|. If we take a single-bond volume Λ = {b}, where b ∈ L, we obtain the marginal From this we get that if ν l = νl then condition (3.25) is fulfilled for all vectors (n 0 , n 1 , . . . , n q−1 ) ∈ {0, 1, 2} q with i∈Zq n i = 2.
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. We may further assume m 1 = 0 and write m = m 2 . Then this equation reduces to a n 1 + 1 a n 2 + 1 = a n−m To prove the other direction we must show that a 1 = a 2 and a 1 a 2 = 1 are the only solutions to the system (3.30): For any x > 0 set f (n,m) (x) := x n +1 x n−m +x m . Then (3.30) is equivalent to f (n,m) (a 1 ) = f (n,m) (a 2 ). Consider any 0 < m < n. Clearly f (n,m) is continuous, strictly decreasing on (0, 1) and strictly increasing on (1, ∞), which means that for any x ∈ (0, 1] there is at most one y ∈ [1, ∞) with f (n,m) (x) = f (n,m) (y). Since f (n,m) (x) = f (n,m) ( 1 x ) we have that f (n,m) (a 1 ) = f (n,m) (a 2 ) if and only if a 1 = a 2 or a 1 a 2 = 0. Lemma 2. Consider any 4-periodic boundary law of the type and denote the associated GGM by ν (a,b) . Let (a 1 , b 1 ), (a 2 , b 2 ) be two such boundary laws. If ν (a 1 ,b 1 ) = ν (a 2 ,b 2 ) then necessarily  (2) gives (3.31) Setting x := a 1 + b 1 and y := a 2 + b 2 leads to the equation x 2 +4 y 2 +4 = x y which is equivalent to (x − y)(xy − 4) = 0. This completes the proof.
c) 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: a) First note that for any subtree of the k-regular tree with n vertices we have n(k + 1) − 2(n − 1) = n(k − 1) + 2 points in the outer boundary which follows by induction on n (see [20]). Thus ν (c 1 ) = ν (c 2 ) if and only if for each n ∈ N the equation (3.25) holds true for all (n 0 , n 1 , n 2 ) ∈ N 0 3 with n 0 +n 1 +n 2 = n(k−1)+2. This is equivalent to the existence of some λ > 0 depending only on k and n with c (n 0 +n 1 ) 1 for all such vectors (n 0 , n 1 , n 2 ).
Remark 3. Lemma 1 can also be concluded from Lemma 2 and the fact that two periodic boundary laws lead to the same GGM if one is obtained from the other by cyclic permutations or multiplication with a positive constant.

Translation-invariant solutions
In this section we calculate periodic solutions to the boundary law equation for the SOS-model. First let β > 0 be any inverse temperature and set θ := exp(−β) < 1. The transfer operator Q then reads Q(i − j) = θ |i−j| for any i, j ∈ Z and a spatially homogeneous boundary law, now denoted by z, is any positive function on Z solving the system (3.17), whose values we will denote by z i instead of z(i). Further notice that a boundary law is only unique up to multiplication with any positive prefactor. Hence we may choose this constant in a way such that we have z 0 = 1. At last set Z 0 := Z \ {0}. Taking into account these prerequisites the boundary law equation (3.17) now reads

4.1.
A simplification of the system (4.1). Let z(θ) = (z i = z i (θ), i ∈ Z 0 ) be a solution to (4.1). Denote It is clear that each l i and r i can be a finite positive number or +∞. Proof. The proof follows from the following equalities In what follows, we will always assume that l 0 < +∞ and r 0 < +∞.
This system can be written as (4.5) Proposition 1. A vector u = (u i , i ∈ Z), with u 0 = 1, is a solution to (4.5) if and only if for u i (= k √ z i ) the following holds where τ = θ −1 + θ = 2 cosh(β).
Sufficiency. Assume (4.6) holds. Then we get (4.7) with some C =C. Write this equality for i replaced by i + 1 − j, i.e.
Multiply both sides of (4.8) by θ j and sum over j = 1, 2, . . . . Here, absolute convergence of all occurring infinite sums is guaranteed by the assumption l 0 < +∞ and r 0 < +∞ and the fact that θ < 1. Then after rearrangement/simplifications we get Dividing both sides of this equality by θ we get Now rewrite (4.7) for i replaced by i + j, C is replaced byC and multiply both sides of the obtained equality by θ j then sum over j = 1, 2, . . . . After simplifications we get Adding (4.9) and (4.10) we get the ith equation of (4.5) with C replaced byC. For u 0 = 1 we getC = C.
Lemma 5. If l 0 < +∞ and r 0 < +∞ then we have Proof. Using (4.6) we get Compute the following Since θ −1 + θ − τ = 0 we get the formula of l 0 . The case r 0 is similar.
By this Lemma we have The equation (4.6) can be separated into the following independent recurrent equations where i ≥ 0, u 0 = 1 and u −1 , u 1 are some initial numbers. Note that for i = 0 the above equations are trivially fulfilled for all values of u 1 and u −1 . Hence it suffices to consider (4.12) and (4.13) for i ≥ 1.

4.2.
A class of 4-Periodic solutions to (4.6). In this subsection we shall describe the two-parameter family of solutions to (4.6) which have the form where a and b some positive numbers. Such a solution defines a periodic two-side infinite sequence, i.e. . .., a, 1, b, 1, a, 1, b, 1, a, 1, b, ... (4.15) The equations (4.12) and (4.13) give the following system of equations For simplicity we consider the case k = 2 and give full analysis of the system (4.16).
In case k = 2 subtracting from the first equation of the system the second one we get Which gives three possibilities: One easily gets the following solutions to this equation (recall that τ > 2): • If τ ≤ 6 then the equation (4.18) has unique solution a 0 = 1.
. In this case similarly as in previous case we obtain (τ + τ 2 − 4τ )a 2 − 2τ a + 4 = 0 which for τ ≥ 2 + 2 √ 5 has the following solutions Using (4.17) we get b 5 = a 6 and b 6 = a 5 . Clearly all of these solutions are positive (see Fig. 2). Taking into account the freedom of cyclic permutations of boundary laws we thus proved the following:  where the values a i and b i are defined above.  We will denote the GGM assigned by Theorem 3 to a boundary law z (a 2 ,b 2 ) n by ν (a 2 ,b 2 ) .
It is well known (see [15], p.28) that the number of positive roots of the polynomial (4.22) does not exceed the number of sign changes of its coefficients. It is obvious that 2 − τ < 0. Thus the number of positive roots of the polynomial (4.22) is at most 2.
The following lemma gives the full analysis of the equation (4.22): 5. if τ = τ 0 , then the equation has two solutions, one of which is a = 1.
We have ψ k (a) > 2, a > 0 and ψ ′ k (a) = 0 is equivalent to The last polynomial equation has exactly one positive solution, because signs of its coefficients changed only one time, and at a = 0 it is negative, i.e. -1 and at a = +∞ it is positive. Denote this unique solution by a * . Then ψ k (a) has unique minimum at a = a * , and lim a→0 ψ k (a) = lim a→+∞ ψ k (a) = +∞ (see Fig.3). Thus τ c = τ c (k) = min a>0 ψ k (a) = ψ k (a * ).
These properties of ψ k (a) completes the proof.
Thus taking into account Lemma 3 and Lemma 6 we obtain the following: