On the uniqueness of Gibbs measure in the Potts model on a Cayley tree with external field
Leonid V. Bogachev, Utkir A. Rozikov
TL;DR
This paper investigates the conditions for the uniqueness of Gibbs measures in the Potts model on Cayley trees with external fields, providing new criteria and phase diagrams for different parameters.
Contribution
It introduces a novel criterion for translation invariance of splitting Gibbs measures and analyzes the phase diagram for the Potts model with external fields on Cayley trees.
Findings
Derived sufficient conditions for uniqueness of Gibbs measures.
Identified phase diagrams on the temperature-field plane for various parameters.
Conjectured a universal bound on the number of homogeneous Gibbs measures.
Abstract
The paper concerns the -state Potts model (i.e., with spin values in ) on a Cayley tree of degree (i.e., with edges emanating from each vertex) in an external (possibly random) field. We construct the so-called splitting Gibbs measures (SGM) using generalized boundary conditions on a sequence of expanding balls, subject to a suitable compatibility criterion. Hence, the problem of existence/uniqueness of SGM is reduced to solvability of the corresponding functional equation on the tree. In particular, we introduce the notion of translation-invariant SGMs and prove a novel criterion of translation invariance. Assuming a ferromagnetic nearest-neighbour spin-spin interaction, we obtain various sufficient conditions for uniqueness. For a model with constant external field, we provide in-depth analysis of uniqueness vs.\ non-uniqueness in theā¦
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On the uniqueness of Gibbs
measure in the Potts model on a Cayley tree with external field
LeonidĀ V.Ā Bogachev
L.āV.āBogachev
Department of Statistics, School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK.
Ā andĀ
UtkirĀ A.Ā Rozikov
U.āA.āRozikov
Institute of Mathematics, 81 Mirzo Ulugābek str., 100170, Tashkent, Uzbekistan.
To the memory of H.-O.Ā Georgii
Abstract.
The paper concerns the -state Potts model (i.e., with spin values in ) on a Cayley tree of degree (i.e., with edges emanating from each vertex) in an external (possibly random) field. We construct the so-called splitting Gibbs measures (SGM) using generalized boundary conditions on a sequence of expanding balls, subject to a suitable compatibility criterion. Hence, the problem of existence/uniqueness of SGM is reduced to solvability of the corresponding functional equation on the tree. In particular, we introduce the notion of translation-invariant SGMs and prove a novel criterion of translation invariance. Assuming a ferromagnetic nearest-neighbour spin-spin interaction, we obtain various sufficient conditions for uniqueness. For a model with constant external field, we provide in-depth analysis of uniqueness vs.Ā non-uniqueness in the subclass of completely homogeneous SGMs by identifying the phase diagrams on the ātemperatureāfieldā plane for different values of the parameters andĀ . In a few particular cases (e.g., or ), the maximal number of completely homogeneous SGMs in this model is shown to be , and we make a conjecture (supported by computer calculations) that this bound is valid for all andĀ .
Key words and phrases:
Cayley tree, configuration, Gibbs measure, generalized boundary conditions, boundary law, random external field
2010 Mathematics Subject Classification:
Primary 82B26; Secondary 60K35
Contents
-
2.3 Translation-invariant SGM and the problem of non-uniqueness
-
5.1 Classification of positive solutions to the systemĀ (2.31)
1. Introduction
1.1. Background and motivation
The Potts model was introduced by R.āB.Ā Potts [43] as a lattice system with spin states and nearest-neighbour interaction, aiming to generalize the KramersāWannier duality [28] of the Ising model (). Since then, it has become the darling of statistical mechanics, both for physicists and mathematicians [4, 64], as one of few āexactly solubleā (or at least tractable) models demonstrating a phase transition [11, 15, 16, 27, 31, 35]. Due to its intuitive appeal to describe multistate systems, combined with a rich structure of inner symmetries, the Potts model has been quickly picked up by a host of research in diverse areas, such as probability [25], algebra [33], graph theory [5], conformally invariant scaling limits [46, 54], computer science [18], statistics [23, 39], biology [24], medicine [58, 59], sociology [53, 55], financial engineering [45, 60], computational algorithms [10, 17], technological processes [52, 62], and many more.
Much of this modelling has involved interacting spin system on graphs. In this context, tree-like graphs are especially attractive for the analysis due to their recursive structure and the lack of circuits. In particular, regular trees (known as Cayley trees or Bethe lattices [6]) have become a standard trial template for various models of statistical physics (see, e.g., [1, 2, 3, 34, 37, 61, 63]), which are interesting in their own right but also provide useful insights into (harder) models in more realistic spaces (such as lattices ) as their āinfinite-dimensionalā approximation [4, ChapterĀ 4]. On the other hand, the use of Cayley trees is often motivated by the applications, such as information flows [38] and reconstruction algorithms on networks [15, 36], DNA strands and Holliday junctions [48], evolution of genetic data and phylogenetics [14], bacterial growth and fire forest models [12], or computational complexity on graphs [18]. Crucially, the criticality in such models is governed by phase transitions in the underlying spin systems.
It should be stressed, however, that the Cayley tree is distinctly different from finite-dimensional lattices, in that the ratio of the number of boundary vertices to the number of interior vertices in a large finite subset of the tree does not vanish in the thermodynamic limit.111This is the common feature of nonamenable graphs (seeĀ [7]). For example, if is the degree of the tree (i.e., each vertex has neighbours), is a āballā of radius (centred at some point ) and is the boundary āsphereā, then
[TABLE]
Therefore, the remote boundary may be expected to have a very strong influence on spins located deep inside the graph, which in turn pinpoints a rich and complex picture of phase transitions, including the number of possible pure phases of the system as a function of temperature.
Mathematical foundations of random fields on Cayley trees were laid by Preston [44] and SpitzerĀ [57], followed by an extensive analysis of Gibbs measures and phase transitions (see Georgii [22, Chapter 12] and Rozikov [47], including historical remarks and further bibliography). The Ising model on a Cayley tree has been studied in most detail (see [47, ChapterĀ 2] for a review). In particular, Bleher et al.Ā [8] described the phase diagram of a ferromagnetic Ising model in the presence of an external random field.222Note that perturbation caused by the field breaks all symmetries of the model, which renders standard arguments inapplicable (cf.Ā [9, Chapter 6]). Using physical argumentation, Peruggi et al.Ā [41, 42] considered the Potts model on a Cayley tree (both ferromagnetic and antiferromagnetic) with a (constant) external field, and discussed the āorder/disorderā transitions (cf.Ā [15, 18]).
In the present paper, we consider a similar (ferromagnetic) model but we are primarily concerned with more general āuniqueness/non-uniquenessā transitions. We choose to work with the so-called splitting Gibbs measures (SGM), which are conveniently defined in the thermodynamic limit using generalized boundary conditions (GBC). To be consistent, permissible GBC fields must satisfy a certain functional equation, which can then be used as a tool to identify the number of solutions. In this approach, it is crucial that any extremal Gibbs measure is SGM, and so the problem of uniqueness is reduced to that in the SGM class.
KĆ¼lske et al.Ā [30] described the full set of completely homogeneous SGMs for the -state Potts model on a Cayley tree with zero external field; in particular, it was shown that, at sufficiently low temperatures, their number is . Recently, KĆ¼lske and Rozikov [29] found some regions for the temperature parameter ensuring that a given completely homogeneous SGM is extreme/non-extreme; in particular, there exists a temperature interval in which there are at least extreme SGMs. In contrast, in the antiferromagnetic Potts model on a tree, a completely homogeneous SGM is unique at all temperatures and for any field (see [47, SectionĀ 5.2.1]).
1.2. Set-up
We start by summarizing the basic concepts for Gibbs measures on a Cayley tree, and also fix some notation.
1.2.1. Cayley tree
Let be a (homogeneous) Cayley tree of degree , that is, an infinite connected cycle-free (undirected) regular graph with each vertex incident to edges.333In the physics literature, an infinite Cayley tree is often referred to as the Bethe lattice, whereas the term āCayley treeā is reserved for rooted trees truncated at a finite height [13, 40]. For example, . Denote by the set of the vertices of the tree and by the set of its (non-oriented) edges connecting pairs of neighbouring vertices. The natural distance on is defined as the number of edges on the unique path connecting vertices . In particular, whenever . A (non-empty) set is called connected if for any the path connecting and lies inĀ . We denote the complement of by and its boundary by \partial\varLambda:=\{x\in\varLambda^{c}\colon\exists\mbox{>!}y\in\varLambda,\ d(x,y)=1\}, and we write . The subset of edges in is denoted .
Fix a vertex x_{\circ}\mbox{;!!}\in V, interpreted as the root of the tree. We say that is a direct successor of if is the penultimate vertex on the unique path leading from the root to the vertex ; that is, and . The set of all direct successors of is denoted . It is convenient to work with the family of the radial subsets centred at , defined for by
[TABLE]
interpreted as the āballā and āsphereā, respectively, of radius centred at the root . Clearly, . Note that if then . In the special case we have . For short, we set .
Remark 1.1*.*
Note that the sequence of balls () is cofinal (see [22, SectionĀ 1.2, pageĀ 17]), that is, any finite subset is contained in some .
1.2.2. The Potts model and Gibbs measures
In the -state Potts model, the spin at each vertex can take values in the set . Thus, the spin configuration on is a function and the set of all configurations is ā . For a subset , we denote by \sigma_{\mbox{;!!}\varLambda}\colon\varLambda\to\varPhi the restriction of configuration to ,
[TABLE]
The Potts model with a nearest-neighbour interaction kernel \{J_{xy}\}_{x,\mbox{>!}y\mbox{>!}\in\mbox{>!}V} (i.e., such that and if ) is defined by the formal Hamiltonian
[TABLE]
where is the Kronecker delta symbol (i.e., if and otherwise), and is the external (possibly random) field. According to (1.1), the spin-spin interaction is activated only when the neighbouring spins are equal, whereas the additive contribution of the external field is provided, at each vertex , by the component of the vector corresponding to the spin value .
For each finite subset () and any fixed subconfiguration (called the configurational boundary condition), the Gibbs distribution \gamma^{\eta}_{\mbox{;!!}\varLambda} is a probability measure in defined by the formula
[TABLE]
where is a parameter (having the meaning of inverse temperature), H_{\mbox{;!!}\varLambda} is the restriction of the Hamiltonian (1.1) to subconfigurations in ,
[TABLE]
and Z^{\eta}_{\mbox{;!!}\varLambda}(\beta) is the normalizing constant (often called the canonical partition function),
[TABLE]
Due to the nearest-neighbour interaction, formula (1.2) can be rewritten as444It is also tacitly assumed in (1.4) that , that is, .
[TABLE]
Finally, a measure \mu=\mu_{\beta,\mbox{>!}\xi} on is called a Gibbs measure if, for any non-empty finite set and any ,
[TABLE]
1.2.3. SGM construction
It is convenient to construct Gibbs measures on the Cayley tree using a version of Gibbs distributions on the balls defined via auxiliary fields encapsulating the interaction with the exterior of the balls. More precisely, for a vector field and each , define a probability measure in by the formula
[TABLE]
where is the normalizing factor and , that is (seeĀ (1.3)),
[TABLE]
The vector field in (1.6) is called generalized boundary conditions (GBC).
We say that the probability distributions (1.6) are compatible (and the intrinsic GBC are permissible) if for each the following identity holds,
[TABLE]
where the symbol stands for concatenation of subconfigurations. A criterion for permissibility of GBC is provided by TheoremĀ 2.1 (see SectionĀ 2.1 below). By Kolmogorovās extension theorem (see, e.g., [56, ChapterĀ II, Ā§ā 3, TheoremĀ 4, pageĀ 167]), the compatibility condition (1.8) ensures that there exists a unique measure \mu^{h}=\mu^{h}_{\beta,\mbox{>!}\xi} on such that, for all ,
[TABLE]
or more explicitly (substitutingĀ (1.6)),
[TABLE]
It is easy to show that so defined is a Gibbs measure (seeĀ (1.5)); since the family is cofinal (see RemarkĀ 1.1), according to a standard result [22, Remark (1.24), pageĀ 17] it suffices to check that, for each and any ,
[TABLE]
where is the Gibbs distribution in with configurational boundary condition (cf.Ā (1.2)). Indeed, denote \omega:=\eta_{\mbox{>!}W_{n+1}\mbox{>!!}}\in\varPhi^{W_{n+1}}\mbox{>!!}, then, due to the nearest-neighbour interaction in the Hamiltonian (1.1) and according to (1.9), we have
[TABLE]
Furthermore, recalling the definitions (1.4) and (1.6), and using the proportionality symbol to indicate omission of factors not depending on , we obtain
[TABLE]
and since both the left- and the right-hand sides of (1.12) are probability measures on \varPhi^{V_{n}}\mbox{;!!}, the relation (1.11) follows.
Definition 1.1**.**
Measure satisfying (1.9) is called a splitting Gibbs measure (SGM).
The term splitting was coined by Rozikov and Suhov [50] to emphasize that, in addition to the Markov property (see [22, SectionĀ 12.1] and also RemarkĀ 1.4 below), such measures enjoy the following factorization property: conditioned on a fixed subconfiguration \sigma_{n}\mbox{>!!}\in\varPhi^{V_{n}}, the values are independent under the lawĀ . Indeed, using (1.6) andĀ (1.9), it is easy to see that, for each and any ,
[TABLE]
where the proportionality symbol indicates omission of factors not depending on , and x^{\prime}\mbox{>!!}=x^{\prime}(x)\in W_{n} is the unique vertex such that .
Remark 1.2*.*
Note that adding a constant to all coordinates of the vector does not change the probability measure (1.6) due to the normalization . The same is true for the external field in the Hamiltonian (1.7). Therefore, without loss of generality we can consider reduced GBC , for example defined as
[TABLE]
The same remark also applies to the external field and its reduced version , defined by
[TABLE]
Of course, such a reduction can equally be done by subtracting any other coordinate,
[TABLE]
Remark 1.3*.*
For , the Potts model is equivalent to the Ising model with redefined spins
[TABLE]
whereby the Hamiltonians in the two models are linked through the relations
[TABLE]
In turn, this leads to rescaling of the inverse temperature \beta=\frac{1}{2}\mbox{;!}\tilde{\beta}.
1.2.4. Boundary laws
Let us comment on the link between the SGM construction outlined in SectionĀ 1.2.3 and an alternative (classical) approach to defining Gibbs measures on tree-like graphs (including Cayley trees), as presented in the book by GeorgiiĀ [22, ChapterĀ 12]. As was already mentioned in [29, pages 641ā642], the family of permissible GBC defines a boundary law in the sense of [22, DefinitionĀ (12.10)] (see also [65]); that is, for any such that , and for all it holds
[TABLE]
where is an arbitrary constant (not depending on ).
To see this, for any and (so that ), set
[TABLE]
which defines the values of the boundary law on ordered edges pointing to the rootĀ . This definition is consistent, in that the equation (1.13) is satisfied (for such edges) due to the assumed permissibility of the GBC (see TheoremĀ 2.1).
The values on the edges pointing away from the root (i.e., such that ) can be identified inductively (up to proportionality constants) using formulaĀ (1.13). The base of induction is set out by choosing and . Then for all we have z_{i}(v,x_{\circ})=\exp\{\beta\mbox{>!}h_{i}(v)\} (seeĀ (1.14)), and equation (1.13) yields
[TABLE]
which defines () up to an unimportant constant factor. If is already defined for all and , then for and we have
[TABLE]
where is the unique vertex such that . Noting that the values z_{j}(x^{\prime}\mbox{>!!},x) () are already defined by the induction hypothesis and that z_{j}(v,x)=\exp\{\beta\mbox{>!}h_{j}(v)\} for all , formula (1.13) yields (again, up to a proportionality constant), which completes the induction step.
Remark 1.4*.*
As a consequence of the equivalence between (permissible) GBC and boundary laws , it follows from [22, TheoremĀ (12.12), pageĀ 243] that any SGM determines a unique Markov chain (see [22, DefinitionĀ (12.2), pageĀ 239]), and vice versa, each Markov chain defines a unique SGM .
Remark 1.5*.*
It is known that for each the Gibbs measures form a non-empty convex compact set in the space of all probability measures on endowed with the weak topology (see, e.g., [22, ChapterĀ 7]). A measure is called extreme if it cannot be expressed as \frac{1}{2}\mbox{>!}\mu_{1}+\frac{1}{2}\mbox{>!}\mu_{2} for some with . The set of all extreme measures in denoted by is a Choquet simplex, in the sense that any can be represented as , with some probability measure on . The crucial observation, which will be instrumental throughout the paper, is that, by virtue of combining [22, TheoremĀ (12.6)] with RemarkĀ 1.4, any extreme measure is SGM; therefore, the question of uniqueness of the Gibbs measure is reduced to that in the SGM class.
Using the boundary law , formula (1.10) can be extended to more general subsets in . Namely, according to [22, formulaĀ (12.13), pageĀ 243] (adapted to our notation), for any finite connected set (and ),
[TABLE]
where is the normalizing factor, x_{\mbox{;!!}\varLambda} denotes the unique neighbour of belonging to , and
[TABLE]
In particular, if then (combining (1.16) withĀ (1.14))
[TABLE]
To link the general expression (1.15) with formula (1.10) for balls , consider part of the boundary defined as
[TABLE]
In other words, consists of the points such that the corresponding vertex x_{\mbox{;!!}\varLambda}\in\varLambda is closer to the root than itself. In view of the definition (1.14), for we get h^{\dagger}_{i}(x,x_{\mbox{;!!}\varLambda})\equiv h_{i}(x). Clearly, if then , but if then the set is non-empty and, moreover, it contains exactly one vertex, which we denote byĀ . Note that is closer to the root than \check{x}_{\mbox{;!!}\varLambda}\in\varLambda, and in this case h^{\dagger}_{i}(\check{x},\check{x}_{\mbox{;!!}\varLambda}) is only expressible through the GBC via a recursive procedure, as explained above.
Thus, formula (1.15) can be represented more explicitly as follows,
[TABLE]
In fact, the last sum in (1.19) includes at most one term, which corresponds to ; more precisely, the latter sum is vacuous whenever , in which case the first sum in (1.19) is reduced to the sum over all . In particular, the formula (1.10) is consistent with (1.19) by picking the set (), with boundary . For a graphical illustration of the sets involved in formula (1.19), see FigureĀ 1 (for a single-vertex set with ).
1.2.5. Layout
The rest of the paper is organized as follows (cf.Ā the table of contents). We state our main results in SectionĀ 2, starting with a general compatibility criterion (TheoremĀ 2.1), which reduces the existence of SGM to the solvability of an infinite system of non-linear equations for permissible GBC . This is followed by various sufficient conditions for uniqueness of SGM with uniform ferromagnetic interaction (Theorems 2.2, 2.3 andĀ 2.5). As part of our general treatment of the Potts model on the Cayley tree, in SectionĀ 2.3.1 we introduce the notion of translation-invariant SGMs (based on a bijection between and a free group with generators of period each), and state a novel criterion of translation invariance (PropositionĀ 2.6) in terms of the external field and the GBC.
Non-uniqueness results for a subclass of completely homogeneous SGMs (i.e., where the reduced fields and are constant) are summarized in Theorems 2.8, 2.9 andĀ 2.10. The number of such measures is estimated in several special cases by (TheoremĀ 2.11), and we conjecture that this is a universal upper bound. In SectionĀ 3, we record some auxiliary lemmas. The proofs of the uniqueness results (Theorems 2.2, 2.3 andĀ 2.5) are presented in SectionĀ 4. SectionĀ 5 is devoted to the in-depth analysis of completely homogeneous SGMs, culminating in the proof of TheoremsĀ 2.8ā2.11 (given in SectionsĀ 5.2ā5.5, respectively). In SectionĀ 6, we study some fine properties (such as monotonicity, bounds and zeros) of the critical curves on the temperatureāfield plane, summarized in Propositions 6.4, 6.5, 6.11ā6.14, 6.16 andĀ 6.17. Finally, AppendixĀ A presents the proof of PropositionĀ 2.6, while AppendixĀ B is devoted to the proof of a technical LemmaĀ 5.3 addressing the special case .
2. Results
2.1. Compatibility criterion
In view of Remark 1.2, when working with vectors and vector-valued functions and fields it will often be convenient to pass from a generic vector to a āreduced vectorā by setting ().
The following general statement describes a criterion555Earlier versions of this theorem are found in [21, PropositionĀ 1, pageĀ 375] or [47, TheoremĀ 5.1, pageĀ 106]. for the GBC to guarantee compatibility of the measures .
Theorem 2.1**.**
The probability distributions defined in (1.6) are compatible (and the underlying GBC are permissible) if and only if the following vector identity holds
[TABLE]
where , ā,
[TABLE]
and the map is defined for and by the formulas
[TABLE]
Remark 2.1*.*
Likewise, Theorem 2.1 is true for any of the possible reductions (see RemarkĀ 1.2).
Remark 2.2*.*
Note that for any .
Remark 2.3*.*
In view of the link (discussed in SectionĀ 1.2.4) between GBC and boundary laws , the compatibility criterion (2.1) is but a reformulation of the consistency property (1.13) of the boundary law.
By virtue of Theorem 2.1, if the GBC and the external field satisfy the functional equation (2.1) for some then there is a (unique) SGM \mu^{h}_{\beta,\mbox{>!}\xi}.
2.2. Uniqueness results
From now on, we confine ourselves to the case of uniform (ferromagnetic) nearest-neighbour interaction by setting if (and otherwise). It will also be convenient to re-parameterize the model by introducing the new parameter termed activity.
For , consider the function
[TABLE]
which can also be written as
[TABLE]
Noting that
[TABLE]
it is evident that is a decreasing function; in particular, for all
[TABLE]
For brevity, introduce the notation
[TABLE]
and for , consider the equation
[TABLE]
or more explicitly (noting that ),
[TABLE]
The left-hand side of (2.9) is a continuous increasing function of ranging from [math] to , which implies that there is a unique solution of the equation (2.9), denoted . In particular, for we get
[TABLE]
Let us also consider the equation
[TABLE]
which has the unique root
[TABLE]
Noting from (2.7) that, for any ,
[TABLE]
and comparing equations (2.9) and (2.11), it follows that
[TABLE]
where the first inequality is in fact strict unless .
Theorem 2.2**.**
Let be the unique solution of the equationĀ (2.9). Then the Gibbs measure \mu_{\theta,\mbox{>!}\xi} is unique for and any external field .
Remark 2.4*.*
It is known that the Ising model on a Cayley tree with zero external field has a unique Gibbs measure if and only if ā(see [8]); that is to say, is the critical activity of the Ising model. Since , our Theorem 2.2 is sharp in this case. Let be the critical activity for the Potts model; its exact value is known only for the binary tree (), namely Ā [30]. Note that but for , so Theorem 2.2 is not sharp already for , .
For , and any , consider the equation (cf.Ā (2.8))
[TABLE]
where and are defined in (2.4) and (2.7), respectively, and
[TABLE]
It can be shown (see Lemma 3.5) that equation (2.13) has a unique root, \theta^{*}_{\mbox{;!!}\gamma}=\theta^{*}_{\mbox{;!!}\gamma}(k,q). More specifically, if then and equation (2.13) is reduced to equation (2.8) (with ), so that (seeĀ (2.10)). However, if then the root \theta^{*}_{\mbox{;!!}\gamma} is an increasing function of parameter with the asymptotic bounds
[TABLE]
Definition 2.1**.**
Given the external field (), define the asymptotic āgapā between its coordinates as follows,
[TABLE]
where
[TABLE]
Theorem 2.3**.**
The Gibbs measure \mu_{\beta,\mbox{>!}\xi} is unique for any \beta\in\bigl{(}0,\ln\theta^{*}_{\mbox{>!!}\varDelta^{\xi}-k}\bigr{)}, where denotes the unique solution of the equation (2.13).
Remark 2.5*.*
As already mentioned, if then and we recover Theorem 2.2 in this case. But if then \theta_{\mbox{>!!}\mbox{>!}\gamma}^{*}(k,q)>\theta_{0}(k,q) for any (seeĀ (2.15)), so that Theorem 2.3 ensures the uniqueness of the SGM \mu_{\beta,\mbox{>!}\xi} on a wider interval of temperatures as compared to TheoremĀ 2.2, for any . Moreover, due to the monotonicity of the map \gamma\mapsto\theta_{\mbox{>!!}\mbox{>!}\gamma}^{*} (Lemma 3.5), a larger gap facilitates uniqueness of SGM; however, the domain of guaranteed uniqueness (in parameter ) is bounded in all cases (seeĀ (2.15)) by .
Remark 2.6*.*
If the external field is random then the gap (2.16) is a random variable measurable with respect to the ātailā -algebra . Intuitively, this means that does not depend on the values of the field on any finite set . If the values of are assumed to be independent (not necessarily identically distributed) for different then, by Kolmogorovās zeroāone law, (and therefore \theta^{*}_{\mbox{>!!}\varDelta^{\xi}-k}=\mathrm{const}) almost surely (a.s.).
Example 2.1*.*
Let us compute the asymptotic gap in a few examples.
- (a)
Let the random vectors () be mutually independent, with independent and identically distributed (i.i.d.)Ā coordinates (, each taking the values with probabilities . Note that ā() and
[TABLE]
The BorelāCantelli lemma then implies that a.s.Ā and hence, according to (2.16), a.s.
- (b)
In the previous example, let us remove the i.i.d.Ā assumption for the coordinates, and instead suppose that each of the random vectors (still mutually independent for different ) can take two values, , with probability each. Then it is clear that (), hence a.s.
- (c)
Extending example (b), suppose that, with some ,
[TABLE]
Of course, for we have and hence ; thus, let . If then it is straightforward to see that
[TABLE]
For , if then, similarly,
[TABLE]
whereas if then
[TABLE]
Thus, in all cases, the BorelāCantelli lemma yields that a.s.Ā (which also includes the case ).
- (d)
Consider i.i.d.Ā vectors () with i.i.d.Ā coordinates (, each with the uniform distribution on . Note that () and, for any ,
[TABLE]
The BorelāCantelli lemma then implies that a.s., and since is arbitrary, it follows that a.s., for each ; hence, according to (2.16), a.s.
- (e)
For a ānon-ergodicā type of example leading to a random gap , suppose that (), where the distribution of is as in exampleĀ (a). That is to say, the values of the field are obtained by duplicating its (random) value at the root. Then, similarly to (2.18), we compute
[TABLE]
- (f)
Finally, the simplest ācoordinate-orientedā choice , , with a fixed , exemplifies translation-invariant (non-random) external fields, including the case of zero field, . Our results for this model will be stated in SectionĀ 2.3; for now, let us calculate the value of the gap . Again, for we have and hence ; thus, let . If then , , hence it is easy to see that . For , if then and (), whereas if then still but (); as a result, for and for .
The following general assertion summarizes Example 2.1. Recall that random variables are said to be exchangeable if the distribution of the random vector is invariant with respect to permutations of the coordinates. The support of (the distribution of) a random variable is defined as the (closed) set comprising all points such that for any we have .
Proposition 2.4**.**
Suppose that the random vectors are i.i.d., and for each their coordinates are exchangeable. Then
[TABLE]
In particular, a.s., unless a.s., in which case a.s.
Proof.
Observe that, by exchangeability of , the distribution of does not depend on and, moreover,
[TABLE]
Denote . From (2.21), it follows that a.s., and therefore, according to (2.16), a.s. On the other hand, for any we have
[TABLE]
and the BorelāCantelli lemma implies that a.s., so that a.s. As a result, a.s., as claimed. ā
Theorem 2.5**.**
Assume that the random external field is as in PropositionĀ 2.4. Let be the root of the equation666The left-hand side of (2.22) does not depend on due to the i.i.d.Ā assumption on .
[TABLE]
where \check{\xi}_{(1)}(x)\equiv{}_{q}\mbox{>!}\check{\xi}_{(1)}(x)=\min_{i\neq q}\bigl{(}\xi_{i}(x)-\xi_{q}(x)\bigr{)} (cf.Ā (2.17)) and the notation is introduced inĀ (2.14). Then, for each and for -almost all realizations of the random field , there is a unique Gibbs measure \mu_{\theta,\mbox{>!}\xi}.
Remark 2.7*.*
Note that Theorem 2.3 guarantees uniqueness of the Gibbs measure in the interval 1<\theta<\theta^{*}_{\mbox{>!!}\varDelta^{\xi}-k}, where \theta^{*}_{\mbox{>!!}\varDelta^{\xi}-k} is the solution of the equation
[TABLE]
By Proposition 2.4, we have (a.s.), and moreover, with positive probability, unless a.s. Thus, excluding the case where , by monotonicity of the function we conclude that \theta^{*}_{\mbox{>!!}\varDelta^{\xi}-k}(k,q)<\theta^{{\dagger}}(k,q), and therefore the domain of uniqueness in Theorem 2.5 is wider than that in TheoremĀ 2.3.
Example 2.2*.*
Let us illustrate Theorem 2.5 with a simple model described in Example 2.1(c), assuming that . Suppose first that . Then, according to the distribution (2.19) and notation (2.14), equation (2.22) specializes to
[TABLE]
By monotonicity of the function , it is clear that the root of this equation is strictly bigger than the root \theta^{*}_{\mbox{>!!}\varDelta^{\xi}-k}=\theta^{*}_{-\alpha-k} of the equation
[TABLE]
in accordance with Remark 2.7. Similarly, if then the distribution (2.19) is replaced by (2.20) and equation (2.22) takes the form
[TABLE]
and by the monotonicity argument it is evident that its root is strictly bigger than the root \theta^{*}_{\mbox{>!!}\varDelta^{\xi}-k}=\theta^{*}_{\alpha-k} of the equation
[TABLE]
again confirming the observation of RemarkĀ 2.7.
2.3. Translation-invariant SGM and the problem of
non-uniqueness
2.3.1. Translation invariance
To introduce the notion of translations on the Cayley tree ā , let be the free group with generators of order each (i.e., ). It is easy to see (cf.Ā [20] and also [47, SectionĀ 2.2]) that the Cayley tree is in a one-to-one correspondence with the group . Namely, start by associating the root with the identity element , and identify the elements with the nearest neighbours of (i.e., comprising the set ). Proceed inductively by expanding the elements along the tree via right-multiplication by the generators ), yielding new elements777Indeed, if a=w\mbox{>!}a_{j} then a\mbox{>!}a_{j}=w\mbox{>!}a_{j}^{2}=w, so this particular multiplication returns the element already obtained at the previous step. corresponding to the set of direct successors of (see FigureĀ 2). This establishes a bijection .
Consider the family of left shifts () defined by
[TABLE]
By virtue of the bijection , this determines conjugate translations on ,
[TABLE]
Clearly, is an automorphism of preserving the nearest-neighbour relation; indeed, if and (so that \mathfrak{b}(y)=\mathfrak{b}(x)\mbox{;!}a_{j}, with some generator ) then, according to (2.23), x^{\prime}\mbox{>!!}:=\widetilde{T}_{z}(x) and y^{\prime}\mbox{>!!}:=\widetilde{T}_{z}(y)=\mathfrak{b}^{-1}(\mathfrak{b}(z)\mbox{>!}\mathfrak{b}(y))=\mathfrak{b}^{-1}(\mathfrak{b}(x^{\prime})\mbox{;!}a_{j}) are nearest neighbours, \langle x^{\prime}\mbox{;!!},y^{\prime}\rangle\in E (see FigureĀ 2). For example, if (whereby the Cayley tree is reduced to the integer lattice ), the action of the shift () can be written in closed form,
[TABLE]
In turn, the map (2.23) induces shifts on configurations ,
[TABLE]
Definition 2.2**.**
We say that SGM \mu^{h}=\mu^{h}_{\theta,\mbox{>!}\xi} is translation invariant (with respect to the group of shifts ) if for each the measures and coincide; that is, for any (finite) and any configuration
[TABLE]
Recall that the quantities () defining the boundary law were introduced in SectionĀ 1.2.4.
Proposition 2.6**.**
An SGM \mu^{h}=\mu^{h}_{\theta,\mbox{>!}\xi} is translation invariant under the group of shifts if and only if the following conditions are satisfied.
- (i)
The reduced field is constant over the tree,
[TABLE]
- (ii)
The reduced field is symmetric,
[TABLE]
- (iii)
For any ,
[TABLE]
This result will be proved in AppendixĀ A.
For , denote by x^{\prime}\mbox{>!!} the unique vertex such that . Then, according to (1.17) andĀ (2.28),
[TABLE]
Note that . Thus, there are (vector) values () that determine a translation-invariant SGM . By translations (2.29), these values (which can be pictured as ācoloursā) are propagated to all vertices in a periodic āchessboardā tiling, except the root where the value is calculated separately, according to the compatibility formulaĀ (2.1).
The criterion of translation invariance given by PropositionĀ 2.6 appears to be new. In Georgii [22, CorollaryĀ (12.17)], a version of this result is established (in the language of boundary laws) for completely homogeneous SGM, that is, assuming the invariance under the group of all automorphisms of the tree .888It is worth pointing out that the latter group is generated by the group of (left) shifts and pairwise inversions between vertices [32, Ā§ā 3.5]. Namely, we have the following
Corollary 2.7**.**
An SGM \mu^{h}=\mu^{h}_{\theta,\mbox{>!}\xi} is completely homogeneous if and only ifā Ā \check{\boldsymbol{\xi}}(x)\equiv\check{\boldsymbol{\xi}}{}^{0}\mbox{>!!} for all and \check{\boldsymbol{h}}(x)\equiv\check{\boldsymbol{h}}{}^{0}\mbox{>!!} for all .
Remark 2.8*.*
In the existing studies of Gibbs measures on trees (see, e.g., [19, 30]), it is common to use the term ātranslation invariantā (and the abbreviation TISGM) having in mind just completely homogeneous SGM. We prefer to keep the terminological distinction between āsingle-colouredā completely homogeneous GBC \check{\boldsymbol{h}}(x)\equiv\check{\boldsymbol{h}}{}^{0}\mbox{>!!} and āmulti-colouredā translation-invariant GBC as characterized by PropositionĀ 2.6. The latter case is very interesting (especially with regard to uniqueness) but technically more challenging, so it is not addressed here in full generality. However, as we will see below, the subclass of completely homogeneous SGM in the Potts model is very rich in its own right.
2.3.2. Analysis of uniqueness
For the rest of SectionĀ 2.3, we deal with completely homogeneous SGM \mu^{h}=\mu^{h}_{\theta,\mbox{>!}\xi}, that is, with the external field and GBC satisfying the homogeneity conditions of CorollaryĀ 2.7. For simplicity, we confine ourselves to the case where all coordinates of the (reduced) vector \check{\boldsymbol{\xi}}{}^{0}\mbox{>!!} are zero except one; due to permutational symmetry, we may assume, without loss of generality, that and ,
[TABLE]
We also write
[TABLE]
Then, denoting z_{i}:=\theta^{\mbox{;!}\check{h}_{i}^{0}/k} (), the compatibility equations (2.1) are equivalently rewritten in the form
[TABLE]
Solvability of the system (2.31) can be analysed in some detail; in particular, we are able to characterize the uniqueness of its solution, which in turn gives a criterion of the uniqueness of completely homogeneous SGM in the Potts model.
The case is trivial, as the system (2.31) will then have the unique solution . The case and has been studied in [29]; these results can be reproduced directly by the methods developed in the present work similarly to a more general (and difficult) case , and are also obtainable in the limit as (see LemmaĀ 5.1(b) and RemarkĀ 5.1).
Thus, let us focus on the new case . By LemmaĀ 5.1(c), the system (2.31) with is reduced either to a single equation
[TABLE]
or to the system of equations (indexed by )
[TABLE]
subject to the condition
[TABLE]
Let us first address the solvability of the equationĀ (2.32). Denote
[TABLE]
In particular, if then (cf.Ā (2.10)). Let us also set
[TABLE]
Clearly, and for . Furthermore, comparing (2.35) and (2.36) observe that and for . For , denote by the roots of the quadratic equation
[TABLE]
with discriminant
[TABLE]
that is,999Here and in what follows, formulas involving the symbols and/or combine the two cases corresponding to the choice of either the upper or lower sign throughout.
[TABLE]
Furthermore, introduce the notation
[TABLE]
Of course, , and one can also show that for all (see the proof of TheoremĀ 2.8 in SectionĀ 5.2). Finally, denote
[TABLE]
so that and for .
Theorem 2.8**.**
Let denote the number of solutions of the equationĀ (2.32). Then
[TABLE]
where is given in (2.35) and are defined byĀ (2.41).
Let us now state our results on the solvability of the set of equations (2.33). For each , consider the functions
[TABLE]
It can be checked (see LemmaĀ 5.2) that for any there is a unique value such that
[TABLE]
and moreover, the function is strictly increasing. Denote by the (unique) value of such that
[TABLE]
Thus, for any the range of the functions and includes positive values,
[TABLE]
and, therefore, the function
[TABLE]
is well defined, where
[TABLE]
Example 2.3*.*
In the case , from (2.42) and (2.44) we obtain explicitly
[TABLE]
In particular, for , and whenever . Comparing (2.35) and (2.47), we also find that if and only if . For example, for we have , , \theta_{\rm c}=\frac{1}{2}\mbox{>!}(\sqrt{153}-3)\doteq 4.6847, that is, , whereas for and we compute and \theta_{1}=1+2\sqrt{5}\doteq 5.4721>\theta_{\rm c}=\frac{1}{2}\mbox{>!}(\sqrt{241}-5)\doteq 5.2621, respectively. Another simple case is (with and any ); indeed, it is easy to see that the condition (2.44) is satisfied with (cf.Ā LemmaĀ 5.2(c)), whence we readily find .
Theorem 2.9**.**
For each , let denote the number of positive solutions of the systemĀ (2.33). Then if and only if and .
As will be shown in PropositionĀ 6.16(b), has a unique zero given by , whereas has a unique zero , which coincides with the zero \theta_{1}^{\mbox{>!}0} ofĀ . It also holds that is a majorant of the family of functions (see PropositionsĀ 6.11 andĀ 6.12).
2.3.3. Uniqueness of completely homogeneous SGM
In the case , there appears to be an additional critical value (see LemmaĀ 5.3)
[TABLE]
For example,
[TABLE]
For , consider the following subsets of the half-plane ,
[TABLE]
Denote the total number of positive solutions of the system of equationsĀ (2.31) by (, ); of course, this number also depends on andĀ . Theorems 2.8 and 2.9 can now be summarized as follows.
Theorem 2.10** (Non-uniqueness).**
**
- (a)
If then if and only if .
- (b)
If then if . TheĀ āonly ifāĀ statement holds true at least for .
- (c)
If then if and only if .
Remark 2.9*.*
The special case in the definition (2.50) and in TheoremĀ 2.10 emerges because for and , there is a (hypothetically unique) solution (u,v)=\bigl{(}\theta-\frac{\theta+2}{k},1\bigr{)} of the systemĀ (2.33), which is, however, not admissible due to the constraintĀ (2.34) and, therefore, does not destroy the uniqueness of solution toĀ (2.31). This hypothesis is conjectured below; if it is true then āifā in TheoremĀ 2.10(b) (i.e., ) can be enhanced to āif and only ifā for all .
Conjecture 2.1**.**
If , and then (u,v)=\bigl{(}\theta-\frac{\theta+2}{k},1\bigr{)} is the sole solution of the systemĀ (2.33).
Remark 2.10*.*
Regardless of ConjectureĀ 2.1, the inclusion of a proper part of the curve in the uniqueness region in the case is indeed necessary. Namely, it can be proved (see PropositionĀ 5.4) that if is small enough then for but for , where is the zero of .
Theorem 2.10 provides a sufficient and (almost) necessary condition for the uniqueness of solution ofĀ (2.31), illustrated in FigureĀ 3 for and , both with .
To conclude this subsection, the following result describes a few cases where it is possible to estimate the maximal number of solutions of the systemĀ (2.31).
Theorem 2.11**.**
**
- (a)
If then ; moreover, for all large enough.
- (b)
Let and . Then for all ; moreover, for all large enough.
- (c)
If then for all and .
Conjecture 2.2**.**
The upper bound in TheoremĀ 2.11 appears to be universal. There is empirical evidence from exploration of many specific cases (using the computing package101010Throughout this paper, we used Maple 18 (Build ID 922027) licensed to the University of Leeds. Maple) to conjecture that TheoremĀ 2.11(c) holds true for all .
2.3.4. Some comments on earlier work
For , when the Potts model is reduced to the Ising model, the result of TheoremĀ 2.11(a) is well known (seeĀ [22, SectionĀ 12.2] or [47, ChapterĀ 2]). The case (i.e., with zero field) is also well studied (see, e.g., [47, SectionĀ 5.2.2.2, PropositionĀ 5.4, pages 114ā115] and [30, TheoremĀ 1, pageĀ 192]), the result of TheoremĀ 2.11(b) can be considered as a corollary of [30, TheoremĀ 1, pageĀ 192]); in particular, it is known that there are \bigl{\lfloor}\frac{1}{2}(q+1)\bigr{\rfloor} critical values of , including \theta_{0}^{-}=\theta_{1}^{\mbox{>!}0} and .
The general case with was first addressed by Peruggi et al.Ā [41] (and continued inĀ [42]) using physical argumentation. In particular, they correctly identified the critical point [42, equationĀ (22), pageĀ 160] (cf.Ā (2.35)) and also suggested an explicit critical boundary in the phase diagram for , defined by the expression [42, equationĀ (21), pageĀ 160] (adapted to our notation)
[TABLE]
Note that this function enjoys a correct value at (i.e., , see formulaĀ (6.4) below), but for all . The corresponding critical value of activity , emerging as zero of , is reported in [42, equationĀ (20), pageĀ 158] as
[TABLE]
In particular, is bigger than the exact critical value \theta_{0}^{-}=\theta_{1}^{\mbox{>!}0}, where the uniqueness breaks down at (see PropositionĀ 6.16). For example, the corresponding numerical values (for and or ) are given by (cf.Ā [42, figureĀ 1, pageĀ 159])
[TABLE]
The critical boundary in the phase diagram for was described in [42, pageĀ 160] only heuristically, as a line ājoiningā the points , and , , and illustrated by a sketch graph in the vicinity of (for and or ).
It should be stressed that the phase transition occurring at these critical boundaries is not of type āuniqueness/non-uniquenessā, with which we are primarily concerned in the present paper, but in fact the so-called āorder/disorderā phase transition. The latter was studied rigorously in a recent paper by Galanis et al.Ā [18] in connection with the computational complexity of approximating the partition function of the Potts model. The useful classification of critical points deployed in [18] is based on the notion of dominant phase; in particular, the critical point (conjectured earlier by HƤggstrƶm [26] in a more general context of random cluster measures on trees) can be explained from this point of view as a threshold beyond which only ordered phases are dominant. Note that the paper [18] studies the Potts model primarily with zero external field (); the authors claim that their methods should also work in a more general ferromagnetic framework including a non-zero field, but no details are spelled out clearly.
In the present paper, we do not investigate the thermodynamical nature of phase transitions, instead focussing on the number of completely homogeneous SGMs, especially on the uniqueness issue. In particular, the order/disorder critical point is not immediately detectable by our methods. It would be interesting to look into these issues for the Potts model with external field, thus extending the results ofĀ [18]. More specifically, our analysis (see PropositionĀ 6.16) shows that the critical point is the signature of the upper critical function at , which has a minimum at . Therefore, it is reasonable to conjecture that the analogue of the interval of activities between the critical points and is the interval , where is the (sole) root of the equation and is the smaller root of the similar equation . However, it is not clear as to what happens in the interval between the roots of the latter equation. The counterpart of this picture for is likely to be simpler, as only the equation is involved. We intend to address these issues in our forthcoming work.
3. Auxiliary lemmas
In this section, we collect a few technical results that will be instrumental in the proofs of the main theorems. We start with an elementary lemma.
Lemma 3.1**.**
For , consider the function
[TABLE]
- (a)
If then is monotone increasing on and
[TABLE]
Similarly, if then is monotone decreasing on and
[TABLE]
- (b)
Furthermore,
[TABLE]
Proof.
(a) Differentiating equation (3.1), we get
[TABLE]
If then, according to (3.4), the function is monotone increasing, and the bounds (3.2) follow by taking the limit as . The case is similar.
(b) By the inequality between the arithmetic and geometric means, the denominator on the right-hand side of (3.4) is bounded below by
[TABLE]
and the result (3.3) follows. ā
Let us define two norms for vector ,
[TABLE]
The next two lemmas give useful estimates for the function defined in (2.3) and for its partial derivatives.
Lemma 3.2**.**
For any , the following uniform estimate holds,
[TABLE]
Proof.
Note that , as a function of , may be represented by formula (3.1) with the coefficients
[TABLE]
where
[TABLE]
Therefore, by the estimates (3.2) we have
[TABLE]
Furthermore, noting that , the two-sided bound (3.8) implies the inequality , and the bound (3.6) follows by taking the maximum over . ā
Recall that the function is defined by (2.4). Denote by the gradient of the map ,
[TABLE]
Recall that the norm \|\mbox{>!}{\cdot}\mbox{>!}\|_{1} is defined in (3.5).
Lemma 3.3**.**
For and any such that , it holds
[TABLE]
where is defined in (2.7) and t_{\gamma}(\theta)=\theta+1+(q-2)\mbox{;!}\theta^{\gamma} (see (2.14)). Moreover, the following uniform estimate holds,
[TABLE]
Proof.
Like in the proof of Lemma 3.2, let us represent by formula (3.1) with the coefficients (3.7). Then by Lemma 3.1
[TABLE]
where s=\sum_{j\neq i}\mathrm{e}^{u_{j}}\geq(q-2)\mbox{;!}\theta^{\gamma}=t_{\gamma}(\theta)-\theta-1. Hence, by monotonicity of the map , from (3.11) it follows that
[TABLE]
On the other hand, expressing by formula (3.1) with
[TABLE]
by Lemma 3.1 we obtain
[TABLE]
If then the estimate (3.13) specializes to
[TABLE]
Similarly, if then and from (3.13) we obtain
[TABLE]
Thus, combining (3.14) and (3.15), we have
[TABLE]
As a result, according to the definition (3.5) of the norm , the bounds (3.12) and (3.16) imply the estimateĀ (3.9).
Finally, since
[TABLE]
from (3.9) we obtain (3.10), and the proof of LemmaĀ 3.3 is complete. ā
Remark 3.1*.*
Estimates similar to (3.9) were proved in [49].
Lemma 3.4**.**
For integer and any , the map is a continuous, strictly increasing function with the range .
Proof.
Denoting , by formula (2.5) we have
[TABLE]
Clearly, the function (3.18) is continuous, so we only need to show that
[TABLE]
is an increasing function.
First of all, if then (3.19) is reduced to , so there is nothing to prove. Suppose that . Differentiating (3.19), it is easy to see that
[TABLE]
and it is evident that the right-hand side of (3.20) is positive for all as long as . On the other hand, for by Bernoulliās inequality we have
[TABLE]
and the expression in square brackets in (3.20) is estimated from below by (\theta+2)+1+(\theta-1)=2\mbox{>!}\theta+2>0. Thus, in all cases for , as required.
Finally, from (3.19) we see that and , and it follows that the range of the function (3.18) is , which completes the proof of the lemma. ā
Lemma 3.5**.**
For , and any , the equationĀ (2.13) has a unique root \theta^{*}_{\mbox{;!!}\gamma}=\theta^{*}_{\mbox{;!!}\gamma}(k,q). If then \theta^{*}_{\mbox{;!!}\gamma}(k,2)\equiv\theta_{0}(k,2)=(k+1)/(k-1), where is the root of the equation (2.9). For , the root \theta^{*}_{\mbox{;!!}\gamma} is a continuous monotone increasing function of , such that
[TABLE]
where is the root of the equation (2.12);* in particular,*
[TABLE]
Proof.
The case is straightforward, so assume that . Due to continuity and monotonicity of the function (see (2.7)) and by virtue of Lemma 3.4, the left-hand side of equation (2.13) is a continuous increasing function of , with the range because
[TABLE]
Hence, the equation (2.13) always has a unique solution, \theta^{*}_{\mbox{;!!}\gamma}=\theta^{*}_{\mbox{;!!}\gamma}(k,q). Since is a continuous increasing function of , while the map is continuous and decreasing, it follows that the root \theta^{*}_{\mbox{;!!}\gamma} is continuous and increasing in .
Finally, observing that (see (2.6) andĀ (3.17))
[TABLE]
and comparing equation (2.13) with the limiting equations as (which have the roots and , respectively), we obtain the required limits (3.21), and hence the asymptotic bounds (3.22) for \theta^{*}_{\mbox{;!!}\gamma}. ā
4. Proofs of the main results related to uniqueness
4.1. Proof of Theorem 2.1 (criterion of compatibility)
For shorthand, denote temporarily . Suppose that the compatibility condition (1.8) holds. On substituting (1.6), it is easy to see that (1.8) simplifies to
[TABLE]
for any . Consider the equality (4.1) on configurations that coincide everywhere in except at vertex , where and . Taking the log-ratio of the two resulting relations, we obtain
[TABLE]
which is readily reduced to (2.1) in view of the notation (2.2) andĀ (2.3).
Conversely, again using (2.2) andĀ (2.3), equation (2.1) can be rewritten in the coordinate form as follows,
[TABLE]
where (omitting the immaterial dependence on , and ) we denote
[TABLE]
Hence, using (4.2) and setting , we get
[TABLE]
Finally, observe that
[TABLE]
whereas from the right-hand side of (4.3) the same sum is given by
[TABLE]
Hence, and formula (4.3) yields (1.8), as required. This completes the proof of TheoremĀ 2.1.
4.2. Preparatory results for the uniqueness of SGM
First, let us rewrite the functional equation (2.1) in a form more convenient for iterations. Recall that we assume () and use the notation .
Lemma 4.1**.**
Via the substitutions
[TABLE]
and
[TABLE]
equation (2.1) is equivalent to the fixed-point equation
[TABLE]
where the mapping is defined by
[TABLE]
Proof.
By means of (4.4), the recursive equation (2.1) for can be written as (4.5). Substituting this into (4.4) and using the notation (4.7), we see that solves the functional equationĀ (4.6). Conversely, if satisfies the equationĀ (4.6) then for defined by (4.5) we have, using (4.7),
[TABLE]
so that solves the equationĀ (2.1). Thus, LemmaĀ 4.1 is proved. ā
In particular, LemmaĀ 4.1 implies that for the proof of uniqueness of SGM it suffices to show that the equation (4.6) has a unique solution ().
Let us state and prove one general result in the contraction case. On the vector space of -valued functions on the vertex set of the Cayley tree , introduce the -norm
[TABLE]
Sometimes, we need the similar norm for functions restricted to subsets ,
[TABLE]
The next lemma and its proof are an adaptation of a standard result for .
Lemma 4.2**.**
For any subset , the space is complete with respect to the sup-norm (4.8).
Proof.
Let be a Cauchy sequence in , that is, for any there is such that for any we have . In particular, is bounded, for some and all . Note that every coordinate sequence (, ) is also a Cauchy sequence (in ) because, according to (3.5), ; hence, it converges to a limit which we denote . Clearly, and .
Now, passing to the limit as in each inequality , we obtain , which implies that , for all . Since is arbitrary, we conclude that as , and the lemma is proved. ā
We also require the following simple estimate.
Lemma 4.3**.**
Let be a -function and \nabla f(\boldsymbol{u})=\bigl{(}\frac{\partial f(\boldsymbol{u})}{\partial u_{1}},\dots,\frac{\partial f(\boldsymbol{u})}{\partial u_{q-1}}\bigr{)} its gradient. Then, for any ,
[TABLE]
Proof.
Define the function \psi(t):=f(\boldsymbol{v}+t\mbox{>!}(\boldsymbol{w}-\boldsymbol{v})), , then
[TABLE]
whence the estimate (4.9) readily follows. ā
Theorem 4.4**.**
Suppose that, for some ,
[TABLE]
Then, for every realization of the field , the equation (4.6) has a unique solution.
Proof.
Consider a mapping of the space to itself defined by formula (4.7). Solving the functional equation (4.6) is then equivalent to finding a fixed point of , that is, . The lemmaās hypothesis implies that is a contraction on ; indeed, for any functions and each , we obtain, using (4.7) and LemmaĀ 4.3,
[TABLE]
Noting that for the set contains exactly vertices, and recalling condition (4.10) with , it follows that
[TABLE]
Thus, because is a Banach space (LemmaĀ 4.2), the well-known Banach contraction principle (e.g., [51, TheoremĀ 9.23, page 220]) implies that , that is, for all . It remains to notice that the value of the solution at is uniquely determined from formulas (4.6) and (4.7) using the (unique) values outside . This completes the proof of TheoremĀ 4.4. ā
Remark 4.1*.*
The unique solution can be obtained by iterations [51]; for example, put and define (), then as (i.e., ).
Remark 4.2*.*
It is straightforward to generalize Theorem 4.4 to the case where the vector \beta\mbox{>!}\boldsymbol{\check{\xi}}(x)+\sum_{y\in S(x)}\boldsymbol{g}(y) (seeĀ (4.7)) is guaranteed to be in a convex domain for any function from a suitable subspace , such that is closed with respect to the norm \|\mbox{;!}{\cdot}\mbox{;!}\|_{V} and ā . In that case, the supremum in (4.10) should be taken over all ,
[TABLE]
and the unique solution automatically belongs toĀ \mathscr{D}\mbox{>!!}. For our purposes, it will suffice to consider the balls B(x)=\{\boldsymbol{u}\in\mathbb{R}^{q-1}\colon\|\boldsymbol{u}-\beta\mbox{>!}\boldsymbol{\check{\xi}}(x)\|_{\infty}\leq k\ln\theta\} and the corresponding subspace (see LemmaĀ 3.2).
4.3. Proofs of Theorems 2.2, 2.3 and 2.5 (uniqueness)
By virtue of Remark 1.5, for the uniqueness in the class of all Gibbs measure it suffices to prove it for SGMs.
4.3.1. Proof of Theorem 2.2
By virtue of the uniform bound (3.10) of Lemma 3.3, for every we have
[TABLE]
and the required result follows by TheoremĀ 4.4.
4.3.2. Proof of Theorem 2.3
In view of equation (2.13) with , we have
[TABLE]
By continuity of the map , inequality (4.11) extends to
[TABLE]
for some small enough.
According to the definition (2.16), there exists an integer such that
[TABLE]
For a specific reduction {}_{\ell\mbox{>!}}\boldsymbol{\check{\xi}}(x)\!\in\!{}_{\ell}\check{\mathbb{R}}^{q}, with components (), denote by () the corresponding functions analogous to that were defined in (2.3) under the standard reduction (i.e., with ). Lemma 3.3 (modified to the case of reduction via the -th coordinate) implies that
[TABLE]
where
[TABLE]
Furthermore, exploiting monotonicity and continuity of the function , we obtain from (4.14)
[TABLE]
with
[TABLE]
Due to the bound (4.13), we have t^{*}_{N}(\theta)\geq t_{\mbox{;!!}\varDelta^{\xi}-\delta-k}(\theta), and by monotonicity of it follows that
[TABLE]
according to the estimate (4.12). Together with (4.15), this implies that (cf.Ā condition (4.10))
[TABLE]
Hence, by an extended version of TheoremĀ 4.4 (see RemarkĀ 4.2), it follows that the solution to the functional equation (4.6) is unique on the subset \{x\in V_{\mbox{;!!}N}^{c}\}. Finally, the values of the solution for are retrieved uniquely by the ābackwardā recursion (4.6) usingĀ (4.7). Thus, the proof of Theorem 2.3 is complete.
4.3.3. Proof of Theorem 2.5
Let \mu_{\beta,\mbox{>!}\xi} and \bar{\mu}_{\beta,\mbox{>!}\xi} be two SGMs determined by the functions and , respectively, each satisfying the functional equationĀ (4.6). Our aim is to show that, under the theoremās hypotheses, , which would imply that \mu_{\beta,\mbox{>!}\xi}=\bar{\mu}_{\beta,\mbox{>!}\xi}. The idea of the proof is to obtain a suitable upper bound on the norm for in terms of for , and to propagate this estimate along the tree. To circumvent cumbersome notation arising from the direct iterations, we will use mathematical induction. Consider the filtration consisting of the sigma-algebras generated by the values of the random field in the sequence of expanding balls ,
[TABLE]
Put
[TABLE]
where the expectation does not depend on due to the i.i.d.Ā assumption on the field . Let us first show that for each () we have the upper bound
[TABLE]
where \mathbb{E}(\cdot\mbox{;!}|\mbox{;!}\mathscr{F}_{n-1}) stands for the conditional expectation.
Fix . The base of induction () is obvious, noting that, due to (4.6), (4.7) and LemmaĀ 3.2,
[TABLE]
Suppose now that the bound (4.17) is true for some , and show that it holds for as well. By Lemma 4.3 we have
[TABLE]
where is the ball of radius centred at \beta\mbox{>!}\boldsymbol{\check{\xi}}(x),
[TABLE]
Recalling that , observe that if then, for each ,
[TABLE]
and hence
[TABLE]
with . Therefore, on applying Lemma 3.3 we have
[TABLE]
Thus, returning to (4.18) we get
[TABLE]
Now, take the conditional expectation \mathbb{E}(\cdot\mbox{;!}|\mbox{;!}\mathscr{F}_{n}) on both sides of (4.19), noting that the factor in front of the sum is a random variable measurable with respect to the sigma-algebra , so it can be pulled out of the expectation (see [56, PropertyĀ K*, pageĀ 216]). This yields
[TABLE]
where in the last inequality we used that and also applied the induction hypothesis to each (seeĀ (4.17)). In turn, using the tower property of conditional expectation (see [56, PropertyĀ H*, pageĀ 216]) with , from (4.20) we obtain
[TABLE]
(see (4.16)). Thus, the induction step is completed and, therefore, the claim (4.17) is true for all . In particular, again using the tower property of conditional expectation, from (4.17) we readily get
[TABLE]
Now, if is the (unique) solution of the equationĀ (2.22), then for all . Hence, taking the limit of (4.21) as gives
[TABLE]
and therefore (a.s.)Ā for any . It remains to notice that this equality extends to by the recursionĀ (4.6).
5. Analysis of the model with constant field
5.1. Classification of positive solutions to the
systemĀ (2.31)
Denote
[TABLE]
so that
[TABLE]
Lemma 5.1**.**
Let be a solution to (2.31), with ().
- (a)
If then is the unique solution.
- (b)
If and then either or there is a non-empty subset , with ranging from to , such that
[TABLE]
where satisfies the equation
[TABLE]
- (c)
If and then:**
- (i)
either and , where satisfies the equation
[TABLE]
and, in particular, ā ;
- (ii)
or, provided that , there is a non-empty subset , with ranging from to , such that
[TABLE]
where and satisfy the set of equations
[TABLE]
and, in particular, and .
Proof.
As a general remark, observe that solves the -th equation of the system (2.31) regardless of all other with .
- (a)
Obvious.
- (b)
In this case, the system (2.31) takes the form
[TABLE]
Suppose that the set is non-empty. By virtue of the identity (5.2), for any equation (5.6) is reduced to
[TABLE]
Because the right-hand side of (5.7) does not depend on and the function is strictly increasing for , it follows that (). Specifically, if then equation (5.6) specializes toĀ (5.3).
- (c)
The proof is similar to partĀ (b). First of all, note that , for otherwise the first equation in (2.31) is not satisfied unless or , either of which is ruled out. Next, if then the first equation for in (2.31) specializes toĀ (5.4), as stated.
Suppose now that , then similarly as above we show that (), and the system (2.31) specializes to equationsĀ (5.5) with and ().
Finally, assuming to the contrary that and comparing the equations inĀ (5.5), we would conclude that , that is, , which is ruled out. Hence, as claimed.
Thus, the proof of Lemma 5.1 is complete. ā
Remark 5.1*.*
It is not hard to check that, in the limit as , case (c) of LemmaĀ 5.1 transforms into caseĀ (b).
5.2. Proof of Theorem
By the substitution
[TABLE]
equation (2.32) can be represented in the form
[TABLE]
with the coefficients (cf.Ā (2.36))
[TABLE]
Equation (5.9) is well known in the theory of Markov chains on the Cayley tree (see, e.g., [44, PropositionĀ 10.7] or [57, pageĀ 389]), and it is easy to analyse the number of its positive solutions. The case is obvious. Assuming , it is straightforward to check that is an increasing function, with and ; also, it has one inflection point x_{0}=\frac{1}{2}\left(b\mbox{;!}(k-1)-(k+1)\right), such that is convex for and concave for (note that only when ). Therefore, the equation (5.9) has at least one and at most three positive solutions. In fact, by fixing and gradually increasing the slope of the ray (), it is evident that there are more than one solutions (i.e., intersections with the graph ) if and only if the equation x\mbox{>!}f^{\prime}(x)=f(x) has at least one solution, each such solution corresponding to the line , with , serving as a tangent to the graph at point . In turn, from (5.9) we compute
[TABLE]
and it readily follows that the condition x\mbox{>!}f^{\prime}(x)=f(x) transcribes as the quadratic equation (2.37), with discriminant given byĀ (2.38). Thus, if , that is, , then the equation (2.37) has two distinct roots 0<x_{-}\mbox{;!!}<x_{+}, corresponding to the ācriticalā values (see (2.39) andĀ (2.40)). Furthermore, using (5.11) it is easy to see that the function is increasing on the interval ; hence, a_{-}\mbox{;!!}<a_{+}.
To summarize, if then the equation (5.9) has a unique solution, whereas if then there are one, two or three solutions according as , or , respectively. Adapting these results to equation (2.32), in view of the second formula in (5.10) the condition is equivalent to , with defined in (2.35). The corresponding critical values of the field parameter are determined by the first formula in (5.10), that is,
[TABLE]
leading to formulaĀ (2.41). This completes the proof of TheoremĀ 2.8.
5.3. Proof of Theorem 2.9
For , denote by m^{\prime}\mbox{;!!}:=q-1-m the āconjugateā index, . Recall the notationĀ (2.42),
[TABLE]
where the polynomial is defined inĀ (5.1).
Lemma 5.2**.**
**
- (a)
For every , there is such that the function is increasing for and decreasing for , thus attaining its unique maximum value at ,
[TABLE]
- (b)
For each , the function defined in (5.14) is continuous and monotone increasing, with . Furthermore, has a unique zero , that is,
[TABLE]
- (c)
The value is the unique positive root of the equation
[TABLE]
In particular, if and if .
Proof.
(a) Differentiating (5.13) with respect to , we get
[TABLE]
It is evident that the function in the parentheses in (5.17) is continuous and monotone decreasing in , with the limiting values as and as . Hence, there is a unique root of the equation , that is,
[TABLE]
and, moreover, for and for . Thus, claim (a) is proved.
(b) Note that the derivative exists by the inverse function theorem applied to equationĀ (5.17). DifferentiatingĀ (5.14) and usingĀ (5.18), we get
[TABLE]
Thus, is continuously differentiable and (strictly) increasing.111111The monotonicity of is obvious without proof, because the function is monotone increasing for each , since . Observe from (5.14) that
[TABLE]
On the other hand, from (5.17) we see that if then, for all ,
[TABLE]
Therefore, by partĀ (a), for such we have , hence, for all ,
[TABLE]
Thus, combining (5.19) and (5.20), it follows that there is a unique root of the equation , which proves claimĀ (b).
(c) Elimination of from the system of equations (5.15) and (5.18) gives for a closed equation,
[TABLE]
which can be rearranged to a more symmetric formĀ (5.16). The uniqueness of the root is obvious, because the left-hand side ofĀ (5.16) is a continuous, increasing function in , with the range from to . Finally, observe that for the left-hand side ofĀ (5.16) is reduced to (m-m^{\prime})\cdot k\mbox{>!}(k-1)/2, which vanishes if and is negative if , so that, respectively, or , as claimed.
Thus, the proof of Lemma 5.2 is complete. ā
Remark 5.2*.*
The statements of Lemma 5.2 including the identity (5.15) are valid with a continuous parameterĀ .
We can now proceed to the proof of TheoremĀ 2.9. Assume that . The second equation in the system (5.5) is reduced to
[TABLE]
which can be rewritten, using the notation (5.13), in the form
[TABLE]
Furthermore, substituting (5.22) and (5.23) into the denominator and numerator, respectively, of the ratio in the first equation of (5.5), we get
[TABLE]
Finally, substituting (5.24) back into (5.23), we obtain the equation
[TABLE]
where (cf.Ā (2.43))
[TABLE]
Conversely, all steps above are reversible, so equations (5.24) and (5.25) imply the systemĀ (5.5).
Note from (5.25) that must satisfy the condition , that is, (seeĀ (2.45)); by Lemma 5.2(b), this is possible if and only if . Moreover, the equationĀ (5.25) has a solution if and only if , with the critical threshold defined inĀ (2.46). This completes the proof of TheoremĀ 2.9.
5.4. Proof of TheoremĀ 2.10
Recall that the critical point was defined inĀ (2.48). If then the only solutions of the compatibility system (2.31) are provided by equation (2.32); therefore, TheoremĀ 2.10(a) readily follows from TheoremĀ 2.8.
More generally (i.e., for ), in order that , either there must be at least two solutions of equationĀ (2.32), that is, (see TheoremĀ 2.8), or, since we always have , there should exist at least one solution of the systemĀ (2.33). By TheoremĀ 2.9, such solutions exist if for some ; since is a majorant of the family (see PropositionĀ 6.12), the latter condition is reduced to , which leads to the inclusion . However, we must ensure that this solution also satisfies the constraint (seeĀ (2.34)). By LemmaĀ 6.10, this is certainly true if , that is, , which proves TheoremĀ 2.10(c).
Finally, TheoremĀ 2.10(b) (for ) readily follows by the next lemma about the maximum of the function over the domain (seeĀ (2.45)).
Lemma 5.3**.**
Let and .
- (a)
For all , we have .
- (b)
Let . If then the function has the unique maximum at , that is, for any .
The proof of the lemma is elementary but tedious, so it is deferred to AppendixĀ B.
Remark 5.3*.*
The maximum of the function , as well as the number of solutions of the equation (5.25) for various values of parameters are illustrated in FigureĀ 4 (for the regular case ) and in FigureĀ 5 (for the special case ).
As mentioned in RemarkĀ 2.10, the case is truly critical with regard to the uniqueness. Recall that denotes the number of positive solutions of the system (2.31); the function is defined inĀ (2.46) and is its zero. The proof of the next proposition relies on some lemmas that will be proved later, in Sections 6.2 andĀ 6.3.
Proposition 5.4**.**
Let and . There exists small enough such that
- (a)
* if ,*
- (b)
* for all .*
Proof.
(a) As shown in the proof of LemmaĀ 5.3 (see AppendixĀ B), \partial^{2}K_{1}/\partial v^{2}|_{v=1}\mbox{;!!}<0 for all \theta\in[\mbox{>!}\theta_{1},\tilde{\theta}_{1}); in particular, \partial^{2}K_{1}(\theta_{1},v)/\partial v^{2}|_{v=1}\mbox{;!!}<0. By LemmaĀ 5.2(c), the set (seeĀ (2.45)) is reduced for to the single point , and . By continuity, it follows that for each \theta\in[\mbox{>!}\theta_{1},\theta_{1}+\varepsilon) and all ; that is, the function is concave on and therefore has a unique maximum located at (remembering that \partial K_{1}/\partial v|_{v=1}\mbox{;!!}=0). But the solution with is not admissible (seeĀ (2.34)), hence , with the only solution of (2.31) coming from equationĀ (2.32).
(b) We have , hence (seeĀ (2.46)) and, by LemmaĀ 6.15(b), also . If is the point where the latter maximum is attained, that is, , then it holds that (see LemmaĀ 6.7 below). Furthermore, according to LemmaĀ 5.2(a), is the unique maximum of the function , and in particular .
From the definition (5.26), it follows that also ,121212By the scaling property (6.15), we also have . and since , LemmaĀ 6.15(a) implies that . Thus, the corresponding solution of the systemĀ (2.33) satisfies the conditionĀ (2.34), and therefore .
By continuity of and , for all \theta\in(\theta_{1}^{0}-\varepsilon,\theta_{1}^{0}\mbox{>!}] (with small enough) we still have that , so the maximum is attained outside . Thus, by the same argument as before, the claim follows. ā
5.5. Proof of Theorem 2.11
(a) First, let . According to LemmaĀ 5.1, the system (2.31) is reduced to the single equation (2.32), and by TheoremĀ 2.8 the number of its solutions is not more than ; furthermore, for and , there are exactly three solutions, so the upper bound is attained.
(b) Let now (and ). Due to LemmaĀ 5.1(b), either or the system (2.32) is reduced to the equationĀ (5.3) indexed by , which can be rewritten as (seeĀ (5.13)). By LemmaĀ 5.2(a), the latter equation has no more than two roots. Hence, considering permutations of the values over the places, it is clear that the total number of solutions to (2.32) is bounded by
[TABLE]
Moreover, for large enough, there will be exactly two roots of each of the equations , because as (see LemmaĀ 5.2(b)). Therefore, the upper bound (5.27) is attained.
(c) Finally, let and . First of all, up to three solutions of the system (2.32) arising from the equationĀ (5.4) are ensured by LemmaĀ 5.1 (see also TheoremĀ 2.8). Other solutions are determined by the system (5.5) indexed by , which in turn depends on the solvability of the equationĀ (5.25). In the case , the latter is a polynomial equation of degreeĀ , and therefore has at most four roots , for eachĀ . The value is then determined uniquely by formula (5.24), and it occupies the first place in the vector . As for the root , it occupies out of the remaining places. Counting the total number of such permutations, we get the upper bound
[TABLE]
as required. This completes the proof of TheoremĀ 2.11.
6. Further properties of the critical curves
and
6.1. Properties of
Lemma 6.1**.**
The quantities , defined in (2.40) for , satisfy the identity
[TABLE]
Proof.
Using (2.40) and noting that (see equationĀ (2.37)), we find
[TABLE]
and formula (6.1) follows. ā
Lemma 6.2**.**
Suppose that . Then the following inequalities hold,
[TABLE]
Proof.
From (5.11), for all we get the upper bound
[TABLE]
noting that, by Bernoulliās inequality,
[TABLE]
Hence, , and the first inequality in (6.2) is proved. The second inequality then readily follows from the identity (6.1). ā
Lemma 6.3**.**
The functions satisfy the following identity,
[TABLE]
where is defined inĀ (2.36). In particular, if then for all .
Proof.
Using (5.12), we obtain
[TABLE]
and the identity (6.3) follows upon substituting formulaĀ (6.1). ā
Proposition 6.4**.**
The functions defined in (2.41) have the following āboundaryāĀ values,
[TABLE]
In particular, if and if .
Proof.
Recall from the proof of TheoremĀ 2.8 (see SectionĀ 5.2) that the critical value corresponds to b(\theta_{\rm c})=\bigl{(}\frac{k+1}{k-1}\bigr{)}^{2}, whereby the quadratic equation (2.37) has the double root
[TABLE]
Hence, using (2.40), we find
[TABLE]
which, together with (5.12), yields formulaĀ (6.4).
If then formula (2.35) gives , and it readily follows from (6.4) that . For , using the relation (5.12) observe that the required inequality is reduced to
[TABLE]
that is,
[TABLE]
Denote
[TABLE]
then (6.6) takes the form
[TABLE]
Furthermore, recalling that and is monotone increasing for (see (2.35) andĀ (2.36)), the inequality (6.7) is equivalent to
[TABLE]
that is,
[TABLE]
which is reduced, upon dividing by and substituting , to
[TABLE]
In fact, it is easy to show that for any . Indeed, since , we have , while
[TABLE]
Thus, inequality (6.8) is verified, which implies that , as argued above.
Let us now proveĀ (6.5). Using the definition of and (see (2.36) and (2.38), respectively), we obtain the following asymptotics as ,
[TABLE]
and
[TABLE]
Hence,
[TABLE]
Using formula (2.40), this yields
[TABLE]
Therefore, from (5.12) we get
[TABLE]
and the limit (6.5) follows. ā
Proposition 6.5**.**
The functions satisfy the following bounds,
[TABLE]
where
[TABLE]
Proof.
Using the relationĀ (5.12), the first inequality inĀ (6.2) and the definition of inĀ (5.10), we get
[TABLE]
which proves the boundĀ (6.9).
Similarly, using the second inequality in (6.2) we have
[TABLE]
Noting from (6.11) that, for ,
[TABLE]
it is easy to see that the right-hand side of (6.12) is bounded above by ; hence, the inequality (6.10) follows. ā
Remark 6.1*.*
Note that for . Of course, if then the upper bound (6.10) holds for all ; however, the ordering between and depends on andĀ . For example, for and
[TABLE]
whereas for and
[TABLE]
In fact, for large the upper bound (6.10) fails near ; indeed, using (2.35) we get
[TABLE]
and, according to (6.4),
[TABLE]
Conjecture 6.1**.**
The function is monotone decreasing for all , whereas is decreasing for and increasing for , with the unique minimum at the critical point
[TABLE]
In the case , we have and, by LemmaĀ 6.3, ; hence, the function should be monotone increasing for all .
This conjecture is supported by computer plots (see FigureĀ 6). Towards a proof, we have been able to characterize the unique zero of and to show rigorously that and (see PropositionĀ 6.16(b)), but the monotonicity properties are more cumbersome to verify.
Remark 6.2*.*
Note that the value (6.13) coincides with a known critical point in the case , above which the solution is unstable (see [47, SectionĀ 5.2.2.2, Proposition 5.4]). Our PropositionĀ 6.16(b-i) explains the emergence of this critical point and its explicit valueĀ (6.13).
6.2. Properties of and
Here and below, we assume that . Recall that is the unique maximum of the function (see LemmaĀ 5.2(a)), and is defined by the relation (5.15), where .
Remark 6.3*.*
All results in this section hold true for a continuous parameterĀ (cf.Ā RemarkĀ 5.2), which is evident by inspection of the proofs.
The next lemma describes the useful scaling properties of the functions and (see (5.13) andĀ (5.26)) under the conjugation .
Lemma 6.6**.**
For each and , the following identities hold for all and ,
[TABLE]
Proof.
Recalling the notation (5.1), note that
[TABLE]
and similarly
[TABLE]
Using (5.13) and (6.16), we can write
[TABLE]
and formula (6.14) follows. Furthermore, using (6.14) and substituting (6.16) and (6.17), we obtain
[TABLE]
which proves formula (6.15). ā
For , let be the point where the function attains its (positive) maximum value, that is, (see LemmaĀ 5.2). The next result provides a strict lower bound for (cf.Ā LemmaĀ 5.2(c) for ).
Lemma 6.7**.**
For all in the range 1\leq m\leq\frac{1}{2}\mbox{>!}(q-1), we have
[TABLE]
Proof.
To the contrary, suppose first that for some . Then, according to LemmaĀ 6.6, we have
[TABLE]
and furthermore,
[TABLE]
since by the hypothesis of the lemma. Combining (6.19) and (6.20), we see that , which contradicts the assumption that is the maximum value of the function .
Assume now that for some . Then
[TABLE]
whence (\theta-1)(k-1)=2\mbox{>!}m. Hence,
[TABLE]
which contradicts the assumption .
Thus, the inequality (6.18) is proved. ā
For , let be the point where the function attains its (positive) maximum value, that is, . Note that (see LemmaĀ 5.2(c)). The importance of the next technical lemma is pinpointed by involvement of the expression p_{k}(v)-(k-1)\mbox{;!}L_{m}(v;\theta) in the partial derivative (see formulaĀ (6.23) below).
Lemma 6.8**.**
Let 1\leq m\leq\frac{1}{2}\mbox{>!}(q-1) and .
- (a)
Let be a critical point of the function , that is, any solution of the equation . Assume that either (i) and , or (ii) . Then
[TABLE]
- (b)
In particular, the inequalityĀ (6.21) holds for , that is,
[TABLE]
Proof.
(a) From the definition (5.26), compute the partial derivative
[TABLE]
with the shorthand notation and . Hence, the condition is reduced to the equality
[TABLE]
If then \partial L_{m}/\partial v|_{v=w}\mbox{;!!}>0, and the required inequalityĀ (6.21) readily follows from equationĀ (6.24) using that . Similarly, if then \partial L_{m}/\partial v|_{v=w}\mbox{;!!}<0 and equationĀ (6.24) implies the inequalityĀ (6.21) provided that . Alternatively, if then, noting that (by LemmaĀ 6.7), we obtain, in agreement withĀ (6.21),
[TABLE]
because the function is strictly increasing and .
Lastly, if then \partial L_{m}/\partial v|_{v=v_{m}}\mbox{;!!}=0 and equationĀ (6.24) implies . Again using LemmaĀ 6.7, similarly to (6.25) we get
[TABLE]
so the inequalityĀ (6.21) holds in this case as well.
(b) Let be the point of maximum of the function . According to part (a), we only have to consider the case where and .
If , let \kern 0.6458pt\hbox{\vbox{\hrule height=0.4pt\kern 1.29167pt\hbox{\kern-1.49994ptw\kern-1.00006pt}}\kern 1.07639pt}>v_{m}>w_{m} be such that L_{m}(\kern 0.6458pt\hbox{\vbox{\hrule height=0.4pt\kern 1.29167pt\hbox{\kern-1.49994ptw\kern-1.00006pt}}\kern 1.07639pt};\theta)=L_{m}(w_{m};\theta), then
[TABLE]
Thus, K_{m}(\kern 0.6458pt\hbox{\vbox{\hrule height=0.4pt\kern 1.29167pt\hbox{\kern-1.49994ptw\kern-1.00006pt}}\kern 1.07639pt};\theta)>K_{m}(w_{m};\theta), which contradicts the assumption that provides the maximum value of the function .
Lastly, suppose that, for some ,
[TABLE]
In view of the definition (5.26), condition (6.28) implies that . Let us prove that in this case we must have , which would then automatically imply the required inequalityĀ (6.22) (cf.Ā (6.25) andĀ (6.26)). To the contrary, assume that . If then, using the definition (5.13) and recalling that , the equation (6.28) is reduced to
[TABLE]
whence we find
[TABLE]
Furthermore, noting that
[TABLE]
and substituting (6.29), from (5.13) we get
[TABLE]
since . Thus, is the left root of the equation . Denote by the right root, that is, and \partial L_{m}/\partial v|_{v=\bar{v}}\mbox{;!!}<0. It follows that (seeĀ (5.26)), so the maximum value of the function is also attained at . Returning to formula (6.23), observe that
[TABLE]
because and, as mentioned above, \partial L_{m}/\partial v|_{v=\bar{v}}\mbox{;!!}<0. But the inequality (6.31) implies that there are points such that , a contradiction. Hence, the case is impossible.
Now, suppose that . Then and, in view of the conditionĀ (6.28), from equation (6.24) it readily follows that \partial L_{m}/\partial v|_{v=w_{m}}\mbox{;!!}=0, that is, is the maximum value of the function . Hence, for all we have
[TABLE]
On the other hand, by (6.24) and monotonicity of ,
[TABLE]
By continuity of the functions and , the inequality (6.33) is preserved for all close enough to :
[TABLE]
Using (6.32) and (6.34), from (6.23) it follows that for such we have . But this means that the function is decreasing in the left vicinity of , and thus cannot be a maximum, in contradiction with our assumption. Thus, we have proved that as required, which completes the proof of LemmaĀ 6.8. ā
The next two lemmas provide useful bounds on . First, there is a simple uniform upper bound.
Lemma 6.9**.**
For all m\in[1,q-2\mbox{>!}],
[TABLE]
Proof.
Observe, using the definition (5.13), that
[TABLE]
and also (cf.Ā (5.17))
[TABLE]
Hence, the point lies to the right of the set (seeĀ (2.45)). But and therefore , as claimed inĀ (6.35). ā
The important lower bound for is established next.
Lemma 6.10**.**
**
- (a)
For all in the range , we have
[TABLE]
- (b)
If m=\frac{1}{2}\mbox{>!}(q-1) then the maximum point (which may not be unique) can be chosen so that .
Proof.
If then, by the inequality (6.22) of LemmaĀ 6.8,
[TABLE]
which implies, due to the monotonicity of , that , in line withĀ (6.36). Thus, it remains to consider the case .
Assume first that for some . Using the definition (5.13) and the value , we have
[TABLE]
and also, recalling formula (6.30),
[TABLE]
Substituting (6.30), (6.37) and (6.38) into (6.23), it is easy to check that the condition \partial K_{m}/\partial v|_{v=1}\mbox{;!!}=0 (seeĀ (6.24)) is reduced to
[TABLE]
Since by assumption, the condition (6.39) is only satisfied if , that is, . Conversely, if (i.e., ) then, by the scaling formulaĀ (6.15) of LemmaĀ 6.6, we have the identity
[TABLE]
which implies that the maximum point can always be chosen so as to satisfy the inequality , which proves partĀ (b) of the lemma.
Finally, let and for some . Denote (see LemmaĀ 5.2 and the definitionĀ (2.45)). We need to distinguish between two subcases, (i) and (ii) , which require a different argumentation.
(i) Assuming first that , we will show that then
[TABLE]
which would contradict the assumption that is the maximum value. By formula (6.15) of LemmaĀ 6.6, we have . Hence, recalling the definition (5.26) of the function , the inequalityĀ (6.41) is reduced to
[TABLE]
Note that (cf.Ā (6.20))
[TABLE]
Hence, for the proof of the inequality (6.42), it suffices to show that the function L\mapsto L\mbox{>!}(p_{k}(w_{m})+L)^{-k} is strictly increasing on the interval . Computing the derivative of this function, we see that the claim holds provided that
[TABLE]
or simply if
[TABLE]
The inequalityĀ (6.43) is easy to prove. Indeed, using the assumption , observe from the definition (5.1) that
[TABLE]
On the other hand, according to the scaling formula (6.14) of LemmaĀ 6.6, we have
[TABLE]
by virtue of the assumption . Now, the required inequality (6.43) readily follows from the estimates (6.44) andĀ (6.45).
Thus, the inequality (6.41) is proved, and therefore the assumptions and are incompatible.
(ii) Assume now that and, as before, . Denote
[TABLE]
that is, the set of all critical points of the function (i.e., satisfying the equationĀ (6.24)) that lie to the right of point . By assumption,
[TABLE]
and our aim is to show that this leads to a contradiction.
Since \partial L_{m}/\partial v|_{v=v_{m}}\mbox{;!!}=0 and , formulaĀ (6.23) implies \partial K_{m}/\partial v|_{v=v_{m}}\mbox{;!!}>0 and, therefore, there is at least one critical point , which then automatically belongs to the set , because by LemmaĀ 6.7. There may also be critical points such that ; for these we may assume, without loss of generality, that , for otherwise we consider the point \kern 0.6458pt\hbox{\vbox{\hrule height=0.4pt\kern 1.29167pt\hbox{\kern-1.49994ptw\kern-1.00006pt}}\kern 1.07639pt}>v_{m}>w such that L_{m}(\kern 0.6458pt\hbox{\vbox{\hrule height=0.4pt\kern 1.29167pt\hbox{\kern-1.49994ptw\kern-1.00006pt}}\kern 1.07639pt};\theta)=L_{m}(w;\theta), and it follows (similarly to the derivation of inequalityĀ (6.27)) that K_{m}(\kern 0.6458pt\hbox{\vbox{\hrule height=0.4pt\kern 1.29167pt\hbox{\kern-1.49994ptw\kern-1.00006pt}}\kern 1.07639pt};\theta)\geq K_{m}(w;\theta), which means that such can be removed from the setĀ without affecting the maximum inĀ (6.46).
Now, the idea is to increase the index . Namely, treating as a continuous parameter (see RemarkĀ 6.3), differentiate the function to obtain
[TABLE]
where we used the condition \partial K_{m}/\partial v|_{v=w_{m}}\mbox{;!!}=0 and the definitions (5.13) andĀ (5.26). Owing to LemmaĀ 6.8(b), the right-hand side of (6.47) is positive and, therefore, the function is monotone increasing as long as and m<m_{0}:=\frac{1}{2}\mbox{>!}(q-1). Likewise, every critical point from the original (finite) set generates a continuously differentiable branch as a function of the increasing variableĀ , and an argument similar to (6.47), now based on LemmaĀ 6.8(a), yields that the corresponding function is monotone decreasing up to .
If for some it occurs that then, by continuity, \partial K_{\widetilde{m}}/\partial v|_{v=1}\mbox{;!!}=0, which implies, as was shown before (seeĀ (6.39)), that and therefore is the maximum value of the function . Moreover, combining the monotonicity properties established above with the hypothetical inequalityĀ (6.46), this implies
[TABLE]
that is,
[TABLE]
But this cannot be true, because there is \kern 0.6458pt\hbox{\vbox{\hrule height=0.4pt\kern 1.29167pt\hbox{\kern-1.49994ptw\kern-1.00006pt}}\kern 1.07639pt}>v_{\widetilde{m}} where L_{\widetilde{m}}(\kern 0.6458pt\hbox{\vbox{\hrule height=0.4pt\kern 1.29167pt\hbox{\kern-1.49994ptw\kern-1.00006pt}}\kern 1.07639pt};\theta)=1, so that K_{\widetilde{m}}(\kern 0.6458pt\hbox{\vbox{\hrule height=0.4pt\kern 1.29167pt\hbox{\kern-1.49994ptw\kern-1.00006pt}}\kern 1.07639pt};\theta)=1 is another maximum and, hence, \kern 0.6458pt\hbox{\vbox{\hrule height=0.4pt\kern 1.29167pt\hbox{\kern-1.49994ptw\kern-1.00006pt}}\kern 1.07639pt}\in W_{\widetilde{m}}, thus contradictingĀ (6.48).
This shows that we can exploit the monotonicity properties with respect to variable up to the final value , so that
[TABLE]
and also
[TABLE]
Combining (6.49) and (6.50) withĀ (6.46), it follows that
[TABLE]
But this is impossible, since and, by the scaling relation (6.15) of LemmaĀ 6.6, (see (6.40)), which implies that the maximum values of the function over and must be the same, in contradiction with the inequalityĀ (6.51).
Thus, the hypothesis (6.46) is false, together with the assumption under case (ii) (i.e., with ). This completes the proof of LemmaĀ 6.10. ā
6.3. Properties of and
Proposition 6.11**.**
For each and , we have
[TABLE]
*Moreover, the functions *(seeĀ (2.46)) satisfy the symmetry relation
[TABLE]
Proof.
Like in LemmaĀ 5.2(c), denote v_{m}^{*}\mbox{;!!}:=v_{m}(\theta_{m}). Observe that satisfies the conjugation property
[TABLE]
where . Indeed, computing the left-hand side of (5.16) for and with replaced by , we get, due to LemmaĀ 5.2(c),
[TABLE]
whence (6.54) follows due to the uniqueness of solution.
Now, using (6.54) and the scaling property (6.14), we have
[TABLE]
according to (5.15), and by the uniqueness of solution to the equation (see LemmaĀ 5.2(b)), the equality (6.52) follows.
Finally, the identity (6.53) is valid due to the definition (2.46) and formulaĀ (6.15). ā
Proposition 6.12**.**
Let , and set m_{0}:=\lfloor\frac{1}{2}\mbox{>!}(q-1)\rfloor. Then for
[TABLE]
Proof.
Treating as a continuous parameter (see RemarkĀ 5.2), differentiate the identity (5.15) to obtain
[TABLE]
Using (5.13) and (5.18), the last identity is reduced to
[TABLE]
which yields
[TABLE]
Recalling that for all m<\frac{1}{2}\mbox{>!}(q-1) (see LemmaĀ 5.2(c)), from (6.57) it follows that for m<\frac{1}{2}\mbox{>!}(q-1). For integer , this transcribes as the inequalityĀ (6.55).
Turning to the proof of (6.56), for a given let be the root of the equation . We shall prove a (stronger) continuous version of the inequality (6.56), namely, that the function (defined for ) is monotone decreasing. As before, denote by the point where the function attains its maximum value, and set . Differentiating the function , we obtain (seeĀ (6.47))
[TABLE]
Now, owing to Lemmas 6.8 andĀ 6.10 (see also RemarkĀ 6.3), the right-hand side of (6.58) is negative for all m\in\bigl{[}m^{*},\frac{1}{2}\mbox{>!}(q-1)\bigr{)} and, therefore, the function is monotone decreasing in the closed interval [m^{*},\frac{1}{2}\mbox{>!}(q-1)]. By the definition (2.46), the same holds for the function , as claimed. ā
Proposition 6.13**.**
For all , the functions defined by formula (2.46) satisfy the upper bound
[TABLE]
Moreover, they have the following āboundaryā values,
[TABLE]
Proof.
Let be the point of maximum of the function , so that . Treating the term in the expression (5.26) as an independent parameter , we can write
[TABLE]
By differentiation, it easy to verify that the maximum on the right-hand side of (6.61) is attained at , hence
[TABLE]
Furthermore, by LemmaĀ 6.9, so that
[TABLE]
Substituting this estimate into the right-hand side of (6.62), we obtain
[TABLE]
and therefore (seeĀ (2.46))
[TABLE]
which proves the bound (6.59). In particular, this implies that
[TABLE]
To obtain a matching lower bound, take a specific value
[TABLE]
then, as ,
[TABLE]
and
[TABLE]
Hence,
[TABLE]
and
[TABLE]
Therefore,
[TABLE]
so that
[TABLE]
Thus, combining (6.63) and (6.64), we obtain the second limit inĀ (6.60).
Finally, we turn to the proof of the first limit inĀ (6.60). By virtue of PropositionĀ 6.11, we may assume that m\leq\frac{1}{2}\mbox{>!}(q-1). Then, by LemmaĀ 6.10, and therefore . Hence, from the definition (5.26) we get
[TABLE]
where . By continuity,
[TABLE]
and it follows from the bound (6.65) that , which implies the first limit inĀ (6.60). Thus, the proof of PropositionĀ 6.13 is complete. ā
Proposition 6.14**.**
For each , the function is monotone increasing for .
Proof.
By virtue of LemmaĀ 6.6, , where ; hence, it suffices to prove the claim for in the range . Using the definitions (5.13) andĀ (5.26), differentiate with respect to to obtain
[TABLE]
on account of the identity \partial K_{m}/\partial v|_{v=w_{m}}\mbox{;!!}\equiv 0. To complete the proof, it remains to notice that the right-hand side of (6.66) is positive due to LemmaĀ 6.8(b). ā
Remark 6.4*.*
The result of PropositionĀ 6.14 is not trivial (unlike the similar statement for , see the footnote in the proof of LemmaĀ 5.2(b)), because, for each , we have that and, therefore, as (see formulaĀ (5.26)).
Conjecture 6.2**.**
For each , the function is monotone increasing.
This conjecture is confirmed by computer plots (see FigureĀ 7) and is easy to prove at least for \theta\leq\theta_{m}^{\mbox{>!}0}, where \theta_{m}^{\mbox{>!}0} is the root of the equation (cf.Ā PropositionĀ 6.16(a)); that is, for \theta\leq\theta_{m}^{\mbox{>!}0}. Indeed, for any \theta\in(\theta_{m},\theta_{m}^{\mbox{>!}0}], we have
[TABLE]
because (PropositionĀ 6.14), whereas \ln K_{m}^{*}(\theta)\leq\ln K_{m}^{*}(\theta_{m}^{\mbox{>!}0})=0.
6.4. Zeros of and
Recall that the functions and are defined in (2.41) and (2.46), respectively. As was observed in numerical examples (see Figure 3 and also Figures 6 andĀ 7), the functions and have the same zero, , whereas at . In this subsection, we give a proof of these observations.
Let us first state and prove a lemma. Recall the notation and .
Lemma 6.15**.**
Let and .
- (a)
For any m\in[1,q-2\mbox{>!}], if for some and then .
- (b)
Let . If for some then . In particular, if and only if .
Proof.
(a) Denoting and using the definitionĀ (5.26), the required inequality can be rewritten as
[TABLE]
and the last inequality holds by monotonicity of , since .
(b) Let be such that ; by LemmaĀ 6.10, . On the other hand, , and by part (a) it follows that . The last claim in part (b) then follows by continuity and monotonicity of both and (see LemmaĀ 5.2(b) and PropositionĀ 6.14, respectively), also recalling that implies (seeĀ (5.26)). ā
Proposition 6.16**.**
Let , and set m_{0}:=\bigl{\lfloor}\tfrac{1}{2}(q-1)\bigr{\rfloor}.
- (a)
For each in the range , the function has a unique zero given by
[TABLE]
where v_{m}^{\mbox{>!}0}>1 is a sole positive root of the equation
[TABLE]
-
(b)
-
(i)
The function has a unique zero given by . Moreover, .
- (ii)
The function has a unique zero , which coincides with the zero \theta_{1}^{\mbox{>!}0} of the function .
- (c)
The zeros \theta_{1}^{\mbox{>!}0},\dots,\theta_{m_{0}}^{\mbox{>!}0} follow in ascending order and are strictly below ,
[TABLE]
Proof.
(a) By the definitionĀ (2.46), the condition means that and hence, by Lemma 6.15(b), . Eliminating from the system of equations , \partial L_{m}(v;\mbox{>!}\theta)/\partial v=0 gives for the root v=v_{m}^{\mbox{>!}0} the equation (cf.Ā (5.21))
[TABLE]
which can be rearranged to the formĀ (6.68). Uniqueness of positive solution v_{m}^{\mbox{>!}0} of the equation (6.68) is obvious, noting that the left-hand side of (6.68) is a continuous, increasing function in , with the range from to . To show that v_{m}^{\mbox{>!}0}>1, it suffices to check that the left-hand side of (6.68) at is negative, which is indeed true since . Expressing from the equation L_{m}(v^{\mbox{>!}0}_{m};\theta)=1, we obtain formulaĀ (6.67).
(b) In the limit , the equation (5.4) always has root , while for , by virtue of the identity (5.1), it is reduced to equation (5.3) with . Using the notation (5.13), the latter equation can be rewritten as , which in turn has up to two (positive) roots (see LemmaĀ 5.2). In total, there are three positive roots, and for this number to reduce to two (which is the condition of belonging to the curves ), either (i) one zero of the function must coincide with , or (ii) the equation must have a double root, thus also satisfying the condition .
In case (i), the condition L_{1}(u;\theta)|_{u=1}\mbox{;!!}=1 transcribes as , which immediately yields the root . According to the substitution (5.8) (with ), the corresponding root of the quadratic equation (2.37) is given by , which appears to be the smaller of the two roots, . Therefore, in view of formulas (2.40) and (2.41), the value is a zero of the function . Indeed, using the definition (2.36) of , the second root of (2.37) is found to be
[TABLE]
as claimed.
In case (ii), according to the proof of part (a), the unique solution of the system , is given by (u,\theta)=(v_{1}^{0},\theta_{1}^{\mbox{>!}0}), where
[TABLE]
and is a sole root of the equation (6.68) with , that is,
[TABLE]
Again by the substitution (5.8) with , the corresponding root of the quadratic equation (2.37) is given by
[TABLE]
and we wish to prove that this is the bigger of the two roots, , which would imply like before that \theta_{1}^{\mbox{>!}0} is a zero of the function . Since the other root of (2.37) equals , with defined in (2.36), our claim is expressed as , that is,
[TABLE]
Furthermore, recalling that \theta_{1}^{\mbox{>!}0}>1 (seeĀ (6.70)), we have
[TABLE]
so for the proof of (6.72) it suffices to show that
[TABLE]
Since the function on the left-hand side of (6.71) is monotone increasing, we only need to check that its value at is non-positive, that is,
[TABLE]
which can be rewritten as
[TABLE]
Now, the latter inequality holds because the left-hand side is obviously monotone increasing as a function of , being equal to [math] at . Hence, (6.73) follows.
Thus, we have proved that and \theta_{0}^{-}=\theta_{1}^{\mbox{>!}0} are the sole roots of the functions and , respectively.
Finally, since the function is known to be positive both at and at infinity (see PropositionĀ 6.4), it readily follows that it has a minimum value [math] at , hence , as claimed in partĀ (b)(i).
(c) The ordering inequalities between in (6.69) readily follow from the monotonicity property (6.56) proved in PropositionĀ 6.55. Thus, it remains to show that \theta_{m}^{\mbox{>!}0}<\theta_{0}^{+}. Observe that the value L_{m}(v;\theta)|_{v=1}\mbox{;!!}=(\theta-1)\mbox{>!}(k-1)-(q-1) does not depend on . Recalling that (see the proof of partĀ (a)), we get that , but
[TABLE]
since . Hence, , and by monotonicity of the function it follows, according to the proof in partĀ (a), that \theta_{m}^{\mbox{>!}0}<\theta_{0}^{+}.
An alternative simple argument is that, as shown in partĀ (a), the maximum value of v\mapsto L_{m}(v;\theta_{m}^{\mbox{>!}0})=1 is attained at , hence L_{m}(1;\theta_{m}^{\mbox{>!}0})<1 and, again by monotonicity, it follows that \theta_{0}^{+}>\theta_{m}^{\mbox{>!}0}. ā
A version of PropositionĀ 6.16(b) for is easy to obtain.
Proposition 6.17**.**
In the case , the functions have a unique zero at , which coincides with . Moreover, .
Proof.
By LemmaĀ 6.3 with , we have for all . Thus, it suffices to consider the function . Treating the index as a continuous variable and taking the limit from the domain as , we see that the unique zero of , given by , converges to . On the other hand, the derivative vanishes at for each , hence its limiting value at is also zero, that is, , as claimed.
This result can also be obtained by a direct calculation. Namely, for the definition (2.36) is reduced to . From equation (2.37) with , it is easy to see that as . Moreover, a simple asymptotic analysis shows that
[TABLE]
Hence, from (2.40) it follows
[TABLE]
Finally, substituting this into (2.41) we obtain
[TABLE]
which implies that , as claimed. ā
Example 6.1*.*
Consider the case . Then the zero of the function specializes to . Furthermore, the equation (6.68) is easily solved to yield , and from (6.67) we readily find
[TABLE]
It is of interest to note, by comparing (6.74) with (2.47), that \theta_{m}^{\mbox{>!}0}=\theta_{m+1} (cf.Ā [30, equationĀ (2.1), pageĀ 192]). Finally, taking in (6.74) gives \theta^{-}_{0}=\theta_{1}^{\mbox{>!}0}=1+2\mbox{;!}\sqrt{q-1}.
Acknowledgements
This research was partially supported by a Royal Society International Exchange Grant IES\R2\181078 (December 2018 ā March 2019). U.A.R.Ā acknowledges support by the Institute for Advanced Study of Aix-Marseille University (IMĆ©RA) under a research residency scheme (FebruaryāJuly 2015), by the London Mathematical Society (LMS Scheme 2 visiting grant, Leeds, February 2016), and by the School of Mathematics, University of Leeds (visiting grant, FebruaryāMarch 2016). Research of L.V.B.Ā was partially supported by a visiting grant from SFBā701 (Bielefeld University, June 2015).
Appendix A Proof of Proposition 2.6
Necessity. Assume that the measure is translation invariant, that is, the condition (2.25) holds. Pick the set to be the unit ball centred at the root , with , where the numbering in is consistent with the bijection , that is, (see SectionĀ 2.3.1). For , the shifted set (where , ) is the unit ball centred at , so that . Consider an arbitrary configuration , with spin values
[TABLE]
Note that if then, according to (2.24),
[TABLE]
Hence, the property (2.25) of translation invariance of specializes as follows,
[TABLE]
Using formulas (1.3) and (1.15), and cancelling the common term \beta\sum_{j=1}^{k+1}J\mbox{>!}\delta_{i_{0},i_{j}}, the equality (A.2) is reduced to
[TABLE]
Varying in (A.3) (whilst keeping all other fixed) shows that the difference does not depend on . More precisely, on subtracting from (A.3) the same equality with , we obtain
[TABLE]
Since (A.4) holds for any , this provesĀ (2.26) in view of the notation (2.2).
Furthermore, by virtue of (2.26) the equality (A.3) becomes
[TABLE]
Similarly, varying the values in (A.5) (one at a time) yields, for each ,
[TABLE]
On the other hand, fix and consider the shift , resulting in
[TABLE]
The latter equality follows by recalling the definition of conjugate translations (seeĀ (2.23)) and noting that z_{j}=\widetilde{T}_{z}(x_{j})=\mathfrak{b}^{-1}(\mathfrak{b}(z)\mbox{>!}\mathfrak{b}(x_{j}))=\mathfrak{b}^{-1}(\mathfrak{b}(z)\mbox{>!}a_{j}) and, therefore,
[TABLE]
because (see SectionĀ 2.3.1). Hence, for the result (A.6) transforms into
[TABLE]
Comparing (A.6) and (A.7), we conclude that , and the claim (2.27) follows. Finally, let and , for some . Observe that
[TABLE]
The first equality in (A.8) is automatic; to check the second one, note that
[TABLE]
That is, \mathfrak{b}(y_{j})=\mathfrak{b}(y)\mbox{>!}\mathfrak{b}(x_{j}) and (A.8) follows. Thus, formula (A.6) applied to the edge gives
[TABLE]
Combined with (A.6), this implies
[TABLE]
and (2.28) follows. This completes the āonly ifā part of the proof.
Sufficiency. Suppose that the conditions (2.26), (2.27) and (2.28) are satisfied. It suffices to verify formula (2.25) for the balls (). For , denote , then and . Furthermore, for set (seeĀ (2.24)). Observe that if then, according to (A.1),
[TABLE]
Hence, recalling (1.3), we have
[TABLE]
where at the last step we used the propertyĀ (2.26). Thus, from formula (1.15) we obtain, omitting factors not depending on ,
[TABLE]
where y_{\mbox{;!!}\varLambda}\mbox{;!!} is the unique neighbour of in . Note that if , with x\in\partial V_{n-1}\mbox{>!!}=W_{n}, then y_{\mbox{;!!}\varLambda}\mbox{;!!}=\widetilde{T}_{z}(x^{\prime}), where is the unique vertex such that . Thus, using (A.9), (2.28) andĀ (2.29), we can write
[TABLE]
Returning to (A.10), this gives \mu^{h}(\sigma_{\mbox{;!!}\bar{\varLambda}}\mbox{;!!}=\varsigma_{z})\propto\mu^{h}(\sigma_{V_{n}}\mbox{;!!}=\varsigma), and since
[TABLE]
it follows that \mu^{h}(\sigma_{\mbox{;!!}\bar{\varLambda}}\mbox{;!!}=\varsigma_{z})=\mu^{h}(\sigma_{V_{n}}\mbox{;!!}=\varsigma), and the proof of the āifā part is complete.
Appendix B Proof of LemmaĀ 5.3
B.1. Proof of part (a)
Denote . For (i.e., with m=m^{\prime}\mbox{;!!}=1), the partial derivative (seeĀ (6.23)) can be represented as
[TABLE]
where L_{1}=t\mbox{>!}p_{k}-v^{k}-1 and L_{1}^{\prime}=\partial L_{1}/\partial v=t\mbox{>!}p^{\prime}_{k}-k\mbox{>!}v^{k-1}.
Note that \partial K_{1}/\partial v|_{v=1}\mbox{;!!}=0 (cf.Ā (6.39)); this follows without calculations from the scaling property (seeĀ (6.15)). More explicitly, using the formulas
[TABLE]
it is easy to see that the terms in (B.1) vanish at . We will also need the formula
[TABLE]
Such expressions can be obtained by successively differentiating (at ) the identity
[TABLE]
To compute the second-order derivative at , we need to differentiate in (B.1) only the factors that vanish at , setting elsewhere. Hence,
[TABLE]
Using (B.2) and (B.3), we find
[TABLE]
Finally, substituting formulas (B.2), (B.3) and (B.5)ā(B.7) into (B.4), after simple manipulations (verified with Maple) we obtain
[TABLE]
The quadratic polynomial in (B.8) has one positive zero ,
[TABLE]
which corresponds to as defined inĀ (2.48). Hence, for , which implies that is a local minimum of the function . This completes the proof of LemmaĀ 5.3(a).
B.2. Proof of part (b)
It suffices to show, for (i.e., ), that is the sole root of the equation . A plausible general scheme of the proof of the latter statement may be as follows.
- (1)
First, the condition (seeĀ (6.23)) is reduced to , where is a polynomial in (and also a quadratic polynomial inĀ ).
- (2)
Since , the quotient is a polynomial in , and we wish to prove that for any t\leq t^{*}\mbox{;!!} and all .
- (3)
According to the proof in SectionĀ B.1, the condition is satisfied whenever ; moreover, the critical value t^{*}\mbox{;!!} is determined by the quadratic equation , which implies that for t\in[\mbox{>!}0,t^{*}].
- (4)
Bearing in mind the invariance of under the map , it should be possible to represent the polynomial in the form
[TABLE]
where and is a polynomial inĀ .
- (5)
The crucial step is to show that, for each t\in[\mbox{>!}0,t^{*}], the function is monotone decreasing.
- (6)
Finally, using steps (3) to (5), for any (i.e., ) we have
[TABLE]
Hence, for all t\in[\mbox{>!}0,t^{*}] and , we get
[TABLE]
as required.
In what follows, we implement this scheme in more detail for the cases . All calculations were done analytically and also verified using Maple.
B.2.1. Case
According to formula (2.49), t^{*}\mbox{;!!}=\frac{1}{2}\bigl{(}\sqrt{41}-1\bigr{)}. The polynomial is found to be
[TABLE]
Hence,
[TABLE]
where . This gives
[TABLE]
which is clearly a decreasing function of for any .
B.2.2. Case
By (2.49), we have . The quotient is explicitly given by
[TABLE]
Thus,
[TABLE]
and it is easy to check that this function is decreasing in for any .
B.2.3. Case
Elementary but tedious calculations yield
[TABLE]
Rearranging under the substitution gives R(v;t)=v^{4}\mbox{>!}\widetilde{R}(y;t) with
[TABLE]
In particular, if then
[TABLE]
for t=\frac{1}{18}\bigl{(}\sqrt{1081}-17\bigr{)}=t^{*} (cf.Ā (2.49)), as it should be.
Finally, we need to verify that the function is decreasing for any t\in[\mbox{>!}0,t^{*}]. Unfortunately (but inevitably), technicalities involved in a purely analytic check become quite substantial; however, using Maple to plot the graph of (B.9), with parameter ranging from [math] to , makes the monotonicity evident.
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