On the uniqueness of Gibbs measure in the Potts model on a Cayley tree with external field
Leonid V. Bogachev Utkir A. Rozikov





















 
 
 


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Potts model on a Cayley tree]On the uniqueness of Gibbs
measure in the Potts model on a Cayley tree with external
field




L.V. Bogachev]
Leonid V. Bogachev
L. V. Bogachev
 Department of Statistics, School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK.
L.V.Bogachev@leeds.ac.uk

U. A. Rozikov]
Utkir A. Rozikov

 U. A. Rozikov
 Institute of Mathematics,
81 Mirzo Ulug'bek str., 100170, Tashkent, Uzbekistan.
rozikovu@yandex.com

To the memory of H.-O. Georgii


The paper concerns the q-state Potts model (i.e., with spin values
in {1,…,q}) on a Cayley tree 𝕋^k of degree k≥
2 (i.e., with k+1 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 the subclass
of completely homogeneous SGMs by identifying the phase diagrams on
the “temperature–field” plane for different values of the
parameters q and k.

In a few particular cases (e.g., q=2 or k=2), the maximal number
of completely homogeneous SGMs in this model is shown to be 2^q-1,
and we make a conjecture (supported by computer calculations) that
this bound is valid for all q≥ 2 and k≥2.




[2010]Primary 82B26; Secondary 60K35





[
    [
    
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§ INTRODUCTION



 §.§ Background and motivation


The Potts model was introduced by

R. B. Potts <cit.> as a lattice system with q≥2 spin
states and nearest-neighbour interaction, aiming


to generalize the Kramers–Wannier duality <cit.> of
the Ising model (q=2). Since then, it has become the darling of
statistical mechanics, both for physicists and mathematicians
<cit.>, as one of few “exactly soluble” (or at least
tractable) models demonstrating a phase transition
<cit.>.
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 <cit.>, algebra <cit.>, graph theory
<cit.>, conformally invariant scaling limits
<cit.>, computer science <cit.>, statistics
<cit.>, biology <cit.>, medicine <cit.>,
sociology <cit.>, financial engineering
<cit.>, computational algorithms <cit.>,
technological processes <cit.>, 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 <cit.>) have
become a standard

trial template for various models of statistical physics (see, e.g.,
<cit.>), which are interesting in their own right but also
provide useful insights into (harder) models in more realistic
spaces (such as lattices ℤ^d) as their
“infinite-dimensional” approximation <cit.>.

On the other hand, the use of Cayley trees is often motivated by the
applications, such as information flows <cit.> and
reconstruction algorithms on networks <cit.>, DNA
strands and Holliday junctions <cit.>, evolution of genetic
data and phylogenetics <cit.>, bacterial growth and fire forest
models <cit.>, or computational complexity on graphs
<cit.>. 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.[This is the common feature of nonamenable
graphs (see <cit.>).] For example, if k≥ 2 is the degree
of the tree (i.e., each vertex has k+1 neighbours), V_n is a
“ball” of radius n (centred at some point x_∘) and
∂ V_n=V_n+1∖ V_n is the boundary “sphere”, then

    |∂ V_n|/|V_n|=(k+1)
    k^n/1+(k+1)(k^n-1)/(k-1)→ k-1≥1,     n→∞.

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 <cit.> and Spitzer <cit.>, followed by an extensive
analysis of Gibbs measures and phase transitions (see Georgii
<cit.> and Rozikov <cit.>, including historical
remarks and further bibliography). The Ising model on a Cayley tree
has been studied in most detail (see <cit.> for a
review). In particular, Bleher et al. <cit.> described the
phase diagram of a ferromagnetic Ising model in the presence of an

external random field.[Note that perturbation caused by the
field breaks all symmetries of the model, which renders standard
arguments inapplicable (cf. <cit.>).] Using physical
argumentation, Peruggi et al. <cit.> considered
the Potts model on a Cayley tree (both ferromagnetic and
antiferromagnetic) with a (constant) external field, and discussed
the “order/disorder” transitions (cf. <cit.>).

In the present paper, we consider a similar (ferromagnetic) model
but we are primarily concerned with more general
“uniqueness/non-uniqueness” 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.

Külske et al. <cit.> described the full set of completely
homogeneous SGMs for the q-state Potts model on a Cayley tree with
zero external field; in particular, it was shown that, at
sufficiently low temperatures, their number is 2^q-1. Recently,
Külske and Rozikov <cit.> 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 2^q-1 + q
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 <cit.>).



 §.§ Set-up
 We start by summarizing the basic concepts for Gibbs
measures on a Cayley tree, and also fix some notation.



  §.§.§ Cayley tree
 Let 𝕋^k be a
(homogeneous) Cayley tree of degree k≥2, that is, an infinite
connected cycle-free (undirected) regular graph with each vertex
incident to k+1 edges.[In the physics literature, an
infinite Cayley tree is often referred to as the Bethe
lattice, whereas the term “Cayley tree” is reserved for rooted
trees truncated at a finite height <cit.>.] For example,
𝕋^1=ℤ. Denote by V={x} the set of the
vertices of the tree and by E={⟨ x,y⟩} the set of
its (non-oriented) edges connecting pairs of neighbouring vertices.
The natural distance d(x,y) on 𝕋^k is defined as the
number of edges on the unique path connecting vertices x,y∈ V.
In particular, ⟨ x,y⟩∈ E whenever d(x,y)=1. A
(non-empty) set ⊂ V is called connected if
for any x,y∈ the path connecting x and y lies
in . We denote the complement of  by
^c:=V∖ and its boundary by
∂:={x∈^c∃
y∈, d(x,y)=1}, and we write
=∪∂. The subset of
edges in  is denoted E_:={⟨
x,y⟩∈ E x,y∈}.

Fix a vertex x_∘∈ V, interpreted as the root of
the tree. We say that y∈ V is a direct successor of x∈
V if x is the penultimate vertex on the unique path leading from
the root x_∘ to the vertex y; that is,
d(x_∘,y)=d(x_∘,x)+1 and d(x,y)=1. The set of all direct
successors of x∈ V is denoted S(x).  It is convenient to work
with the family of the radial subsets centred at x_∘, defined
for n∈ℕ_0:={0}∪ℕ by

    V_n:={x∈ V d(x_∘,x)≤ n},     W_n:={x∈ V
    d(x_∘,x)=n},

interpreted as the “ball” and “sphere”, respectively, of radius
n centred at the root x_∘. Clearly, ∂ V_n=W_n+1.
Note that if x∈ W_n then S(x)={y∈ W_n+1 d(x,y)=1}.
In the special case x=x_∘ we have S(x_∘)=W_1. For short,
we set E_n:=E_V_n.


Note that the sequence of balls (V_n) (n∈ℕ_0) is
cofinal (see <cit.>), that is, any
finite subset ⊂ V is contained in some V_n.




  §.§.§ The Potts model and Gibbs measures

In the q-state Potts model, the spin at each vertex x∈ V can
take values in the set :={1,…,q}. Thus, the spin
configuration on V is a function σ V→ and
the set of all configurations is ^V. For a subset
⊂ V, we denote by σ_→ the restriction of configuration σ to
,

    σ_(x):=σ(x),     x∈.

The Potts model with a nearest-neighbour interaction kernel
{J_xy}_x, y∈ V (i.e., such that
J_xy=J_yx and J_xy=0 if d(x,y)1) is defined by the
formal Hamiltonian

    H(σ)=-∑_⟨ x, y⟩∈ EJ_xyδ_σ(x),σ(y)-∑_x∈ Vξ_σ(x)(x),    σ∈^V,

where δ_ij is the Kronecker delta symbol (i.e.,
δ_ij=1 if i=j and δ_ij=0 otherwise),


and ξ(x)=(ξ_1(x),…,ξ_q(x))∈ℝ^q is the external (possibly
random) field. According to (<ref>), 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 x∈ V, by the component of the vector
ξ(x) corresponding to the spin value σ(x).

For each finite subset ⊂ V
(∅) and any fixed subconfiguration
η∈^^c (called the configurational
boundary condition), the Gibbs distribution
γ^η_ is a probability measure in
^ defined by the formula

    γ^η_(ς)=1/Z^η_(β)exp{-β
    H_(ς)+β∑_x∈∑_y∈_^cJ_xyδ_ς(x),η(y)},    ς∈^,

where β∈(0,∞) is a parameter (having the meaning of
inverse temperature), H_ is the restriction
of the Hamiltonian (<ref>) to subconfigurations in
,

    H_(ς)=-∑_⟨ x, y⟩∈
    E_J_xyδ_ς(x),ς(y)-∑_x∈ξ_ς(x)(x),    ς∈^,

and Z^η_(β) is the normalizing constant
(often called the canonical partition function),

    Z^η_(β)=∑_ς∈^exp{-β
    H_(ς)+β∑_x∈∑_y∈_^cJ_xyδ_ς(x),η(y)}.

Due to the nearest-neighbour interaction, formula (<ref>) can
be rewritten as[It is also tacitly assumed in
(<ref>) that ⟨ x,y⟩∈ E, that is, d(x,y)=1.]

    γ^η_(ς)=1/Z^η_(β)exp{-β
    H_(ς)+β∑_x∈∑_y∈∂J_xyδ_ς(x),η(y)},    ς∈^.

Finally, a measure μ=μ_β,ξ on ^V is
called a Gibbs measure if, for any non-empty finite set
⊂ V and any η∈^^c,

    μ(σ_=ς|σ_^c=η)≡γ_^η(ς),    ς∈^.




  §.§.§ SGM construction

It is convenient to construct Gibbs measures on the Cayley tree
𝕋^k using a version of Gibbs distributions on the balls
(V_n) defined via auxiliary fields encapsulating the interaction
with the exterior of the balls. More precisely, for a vector field
V∋ x↦h(x)=(h_1(x),…, h_q(x))∈ℝ^q and each n∈ℕ_0, define a probability
measure in V_n by the formula

    μ^h_n(σ_n)=1/Z_nexp{-β
    H_n(σ_n)+β∑_x∈ W_nh_σ_n(x)(x)},    σ_n∈^V_n,

where Z_n=Z_n(β, h) is the normalizing factor and
H_n:=H_V_n, that is (see (<ref>)),

    H_n(σ_n)=-∑_⟨ x,y⟩∈ E_nJ_xyδ_σ_n(x),σ_n(y)-∑_x∈
    V_nξ_σ_n(x)(x),    σ_n∈^V_n.

The vector field {h(x)}_x∈ V in (<ref>) is
called generalized boundary conditions (GBC).


We say that the probability distributions (<ref>) are
compatible (and the intrinsic GBC {h(x)} are
permissible) if for each n∈ℕ_0 the following
identity holds,

    ∑_ω∈^W_n+1μ^h_n+1(σ_n∨ω)≡μ^h_n(σ),    σ_n∈^V_n,

where the symbol ∨ stands for concatenation of
subconfigurations. A criterion for permissibility of GBC is provided
by Theorem <ref> (see Section <ref> below). By
Kolmogorov's extension theorem (see, e.g., <cit.>), the compatibility
condition (<ref>) ensures that there exists a unique measure
μ^h=μ^h_β,ξ on ^V such that, for all
n∈ℕ_0,

    μ^h(σ_V_n=σ_n)≡μ^h_n(σ_n),    σ_n∈^V_n,

or more explicitly (substituting (<ref>)),

    μ^h(σ_V_n=σ_n)=1/Z_nexp{-β
    H_n(σ_n)+β∑_x∈ W_nh_σ_n(x)(x)},    σ_n∈^V_n.


It is easy to show that μ^h so defined is a Gibbs measure
(see (<ref>)); since the family (V_n) is cofinal (see
Remark <ref>), according to a standard result
<cit.> it suffices to check that, for
each n∈ℕ_0 and any
η∈^V_n^c,

    μ^h(σ_V_n=σ_n|σ_V_n^c=η)≡γ_n^η(σ_n),    σ_n∈^V_n,

where γ^η_n is the Gibbs distribution in V_n with
configurational boundary condition η (cf. (<ref>)).
Indeed, denote ω:=η_
W_n+1∈^W_n+1, then, due to the
nearest-neighbour interaction in the Hamiltonian (<ref>) and
according to (<ref>), we have

    μ^h(σ_V_n=σ_n|σ_V_n^c=η)
       =μ^h(σ_V_n=σ_n|σ_W_n+1=ω)
       =μ^h(σ_V_n+1=σ_n∨ω)/μ^h(σ_W_n+1=ω)
       =μ^h_n+1(σ_n∨ω)/μ^h_n+1(ω).

Furthermore, recalling the definitions (<ref>) and
(<ref>), and using the proportionality symbol ∝ to
indicate omission of factors not depending on σ_n, we
obtain

    μ^h(σ_V_n=σ_n|σ_V_n^c=η)
       ∝μ^h_n+1(σ_n∨ω)
       ∝exp{-β
    H_n+1(σ_n∨ω)+β∑_x∈
    W_n+1h_ω(x)(x)}
       ∝exp{-β
    H_n(σ_n)+β∑_x∈ W_n∑_y∈ S(x)J_xyδ_σ_n(x),ω(y)}
       ∝γ^η_V_n(σ_n),

and since both the left- and the right-hand sides of (<ref>)
are probability measures on ^V_n, the relation
(<ref>) follows.


Measure μ^h satisfying (<ref>) is called a
splitting Gibbs measure (SGM).


The term splitting was coined by Rozikov and Suhov
<cit.> to emphasize that, in addition to the Markov property
(see <cit.> and also Remark <ref> below),
such measures enjoy the following factorization property:
conditioned on a fixed subconfiguration
σ_n∈^V_n, the values {σ(x)}_x∈
W_n+1 are independent under the law μ^h. Indeed, using
(<ref>) and (<ref>), it is easy to see that, for each
n∈ℕ_0 and any ω∈^W_n+1,

    μ^h(   σ_W_n+1=ω |σ_V_n=σ_n)
       ∝∏_x∈ W_n+1exp{β J_x'xδ_σ_n(x'),ω(x)+βξ_ω(x)(x)+β h_ω(x)(x)}
       ∝∏_x∈ W_n+1μ^h(σ(x)=ω(x)
    |σ_V_n=σ_n),

where the proportionality symbol ∝ indicates omission of
factors not depending on ω, and x'=x'(x)∈ W_n is the
unique vertex such that x∈ S(x').



Note that adding a constant c=c(x) to all coordinates h_i(x) of
the vector h(x) does not change the probability
measure (<ref>) due to the normalization Z_n. The same is true
for the external field ξ(x) in the Hamiltonian
(<ref>). Therefore, without loss of generality we can consider
reduced GBC ȟ(x), for example defined
as

    ȟ_i(x)=h_i(x)-h_q(x),     i=1,…,q-1.

The same remark also applies to the external field
ξ and its reduced version
ξ̌(x), defined by

    ξ̌_i(x):=ξ_i(x)-ξ_q(x),     i=1,…,q-1.

Of course, such a reduction can equally be done by subtracting any
other coordinate,

    _ℓȟ_i(x):=h_i(x)-h_ℓ(x),    _ℓξ̌_i(x):=ξ_i(x)-ξ_ℓ(x)     (iℓ).




For q=2, the Potts model is equivalent to the Ising model with
redefined spins

    σ̃(x):=2σ(x)-3 ∈{-1,1},     x∈ V,

whereby the Hamiltonians in the two models are linked through the
relations

    δ_σ(x),σ(y)=σ̃(x)σ̃(y)+1/2,    ξ_σ(x)(x)=ξ_2(x)-ξ_1(x)/2 σ̃(x)+ξ_1(x)+ξ_2(x)/2.

In turn, this leads to rescaling of the inverse temperature
β=1/2β̃.





  §.§.§ Boundary laws

Let us comment on the link between the SGM construction outlined in
Section <ref> and an alternative (classical) approach to
defining Gibbs measures on tree-like graphs (including Cayley
trees), as presented in the book by Georgii <cit.>.
As was already mentioned in <cit.>, the family of
permissible GBC {h(x)}_x∈ V defines a
boundary law {z(x,y)}_⟨ x,y⟩∈
E in the sense of <cit.> (see also
<cit.>); that is, for any x,y∈ V such that ⟨
x,y⟩∈ E, and for all i∈ it holds

    z_i(x,y)=c(x,y)∏_v∈∂{x}∖{y}∑_j∈
    z_j(v,x)exp{β
    J_xvδ_ij+βξ_i(x)+βξ_j(v)},

where c(x,y)>0 is an arbitrary constant (not depending on
i∈).

To see this, for any y∈ V and x∈ S(y) (so that
d(x_∘,y)=d(x_∘,x)-1), set

    z_i(x,y):=exp{β h_i(x)},     i∈,

which defines the values of the boundary law on ordered edges
⟨ x,y⟩ pointing to the root x_∘. This
definition is consistent, in that the equation (<ref>) is
satisfied (for such edges) due to the assumed permissibility of the
GBC {h(x)}_x∈ V (see Theorem <ref>).

The values z_i(x,y) on the edges ⟨ x,y⟩
pointing away from the root x_∘ (i.e., such that
d(x_∘,y)=d(x_∘,x)+1) can be identified inductively (up to
proportionality constants) using formula (<ref>). The base
of induction is set out by choosing x=x_∘ and y∈∂{x_∘}=S(x_∘). Then for all v∈ S(x_∘)
we have z_i(v,x_∘)=exp{β h_i(v)}
(see (<ref>)), and equation (<ref>) yields

    z_i(x_∘,y)=c(x_∘, y) ∏_v∈ S(x_∘)∖{y}∑_j∈exp{β h_j(v)+β J_x_∘,
    vδ_ij+βξ_i(x_∘)+βξ_j(v)},

which defines z_i(x_∘,y) (i∈) up to an unimportant
constant factor. If z_i(x,y) is already defined for all x∈ V_n
and y∈ S(x), then for x∈ W_n+1 and y∈ S(x) we have

    ∂{x}∖{y}=(S(x)∖{y})∪{x'},

where x'∈ W_n is the unique vertex such that x∈ S(x').
Noting that the values z_j(x',x) (j∈) are already
defined by the induction hypothesis and that
z_j(v,x)=exp{β h_j(v)} for all v∈ S(x), formula
(<ref>) yields z_i(x,y) (again, up to a proportionality
constant), which completes the induction step.


As a consequence of the equivalence between (permissible) GBC
{h(x)}_x∈ V and boundary laws
{z(x,y)}_⟨ x,y⟩∈ E, it follows
from <cit.> that any SGM μ^h
determines a unique Markov chain μ (see
<cit.>), and vice versa, each Markov
chain μ defines a unique SGM μ^h.



It is known that for each β>0 the Gibbs measures form a
non-empty convex compact set 𝒢 in the space of all
probability measures on ^V endowed with the weak topology
(see, e.g., <cit.>). A measure μ∈𝒢 is called extreme if it cannot be expressed as
1/2μ_1+1/2μ_2 for some
μ_1,μ_2∈𝒢 with μ_1μ_2. The set of all
extreme measures in 𝒢 denoted by
𝒢 is a Choquet simplex, in the sense that any
μ∈𝒢 can be represented as μ =
∫_𝒢ν ρ(ν), with some probability
measure ρ on 𝒢. The crucial observation, which
will be instrumental throughout the paper, is that, by
virtue of combining <cit.> with
Remark <ref>, any extreme measure
μ∈𝒢 is SGM; therefore, the question of
uniqueness of the Gibbs measure is reduced to that in the SGM class.

Using the boundary law {z(x,y)}_⟨
x,y⟩∈ E, formula (<ref>) can be extended to
more general subsets in V. Namely, according to
<cit.> (adapted to our
notation), for any finite connected set
∅⊂ V (and
=∪∂),

    μ^h(σ_=ς)=1/Z_exp{-β
    H_(ς)+β∑_x∈∂h^†_ς(x)(x,x_)},    ς∈^,

where Z_=Z_(β,
h) is the normalizing factor, x_
denotes the unique neighbour of x∈∂ belonging
to , and

    h^†_i(x,y):=β^-1ln z_i(x,y),     i∈.

In particular, if x∈ S(y) then (combining (<ref>)
with (<ref>))

    h^†_i(x,y)=h_i(x),     i∈.


To link the general expression (<ref>) with
formula (<ref>) for balls V_n, consider part of the
boundary ∂ defined as

    ∂^↓:=
    {x∈∂ S(x)∩=∅}.

In other words, ∂^↓ consists of the
points x∈∂ such that the corresponding vertex
x_∈ is closer to the root x_∘
than x itself. In view of the definition (<ref>), for
x∈∂^↓ we get
h^†_i(x,x_)≡ h_i(x). Clearly, if
x_∘∈ then
∂^↓=∂, but if
x_∘∉ then the set
∂∖∂^↓ is
non-empty and, moreover, it contains exactly one vertex, which we
denote by x̌. Note that x̌∈∂ is
closer to the root x_∘ than
x̌_∈, and in this case
h^†_i(x̌,x̌_) is only
expressible through the GBC {h(x)} via a recursive
procedure, as explained above.

Thus, formula (<ref>) can be represented more
explicitly as follows,

    μ^h(σ_=ς)
    =1/Z_exp{-β
    H_(ς)+β∑_x∈∂^↓h_ς(x)(x)+β∑_x∈∂∖∂^↓h^†_ς(x)(x,x_)}.

In fact, the last sum in (<ref>) includes
at most one term, which corresponds to x=x̌; more
precisely, the latter sum is vacuous whenever
x_∘∈, in which case the first sum in
(<ref>) is reduced to the sum over all
x∈∂. In particular, the formula
(<ref>) is consistent with
(<ref>) by picking the set
=V_n-1 (n≥1), with boundary
∂=W_n. For a graphical illustration of the sets
involved in formula (<ref>), see
Figure <ref> (for a single-vertex set ={v} with
v x_∘).





  §.§.§ Layout
 The rest of the paper is organized as
follows (cf. the table of contents). We state our main results in
Section <ref>, starting with a general compatibility
criterion (Theorem <ref>), which reduces the existence of SGM
μ^h to the solvability of an infinite system of non-linear
equations for permissible GBC {h(x)}. This is
followed by various sufficient conditions for uniqueness of SGM with
uniform ferromagnetic interaction (Theorems <ref>,
<ref> and <ref>). As part of our general treatment of the
Potts model on the Cayley tree, in Section <ref> we
introduce the notion of translation-invariant SGMs (based on a
bijection between 𝕋^k and a free group with k+1
generators of period 2 each), and state a novel criterion of
translation invariance (Proposition <ref>) 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
{ξ̌(x)} and
{ȟ(x)} are constant) are summarized in
Theorems <ref>, <ref> and <ref>. The number
of such measures is estimated in several special cases by 2^q-1
(Theorem <ref>), and we conjecture that this is a universal
upper bound. In Section <ref>, we record some auxiliary
lemmas. The proofs of the uniqueness results (Theorems <ref>,
<ref> and <ref>) are presented in Section <ref>.
Section <ref> is devoted to the in-depth analysis of
completely homogeneous SGMs, culminating in the proof of
Theorems <ref>–<ref> (given in
Sections <ref>–<ref>, respectively). In
Section <ref>, we study some fine properties (such as
monotonicity, bounds and zeros) of the critical curves on the
temperature–field plane, summarized in Propositions
<ref>, <ref>,
<ref>–<ref>, <ref>
and <ref>. Finally, Appendix <ref> presents
the proof of Proposition <ref>, while Appendix <ref> is
devoted to the proof of a technical Lemma <ref> addressing
the special case q=3.




§ RESULTS



 §.§ Compatibility criterion

In view of Remark <ref>, when working with vectors and
vector-valued functions and fields it will often be convenient to
pass from a generic vector u=(u_1,…,u_q)∈ℝ^q to a “reduced vector”
ǔ=(ǔ_1,…,ǔ_q-1)∈ℝ^q-1 by setting ǔ_i:=u_i-u_q
(i=1,…,q-1).

The following general statement describes a
criterion[Earlier versions of this theorem are found in
<cit.> or <cit.>.] for the GBC {h(x)}_x∈ V to
guarantee compatibility of the measures
{μ^h_n}_n∈ℕ_0.


The probability distributions {μ^h_n}_n∈ℕ_0
defined in (<ref>) are compatible (and the underlying GBC
{h(x)}_x∈ V are permissible) if and only if the
following vector identity holds

    βȟ(x)=∑_y∈
    S(x)F(βȟ(y)+βξ̌(y);
    ^β J_xy),     x∈ V,

where
ȟ(x)=(ȟ_i(x),…,ȟ_q-1(x)),
 ξ̌(x)=(ξ̌_i(x),…,ξ̌_q-1(x)),

    ȟ_i(x):=h_i(x)-h_q(x),     ξ̌_i(x):=ξ_i(x)-ξ_q(x),     i=1,…,q-1,

and the map
F(u;θ)=(F_1(u;θ),…,F_q-1(u;θ))
is defined for
u=(u_1,…,u_q-1)∈ℝ^q-1 and
θ>0 by the formulas

    F_i(u;θ):=ln(θ-1)^u_i+1+∑_j=1^q-1^u_j/θ+
    ∑_j=1^q-1^u_j,     i=1,…, q-1.





Likewise, Theorem <ref> is true for any of the q possible
reductions (see Remark <ref>).



Note that F(0;θ)=0 for
any θ>0.



In view of the link (discussed in Section <ref>) between
GBC {h(x)}_x∈ V and boundary laws
{z(x,y)}_⟨ x,y⟩∈ E, the
compatibility criterion (<ref>) is but a reformulation of the
consistency property (<ref>) of the boundary law.


By virtue of Theorem <ref>, if the GBC {h(x)}
and the external field {ξ(x)} satisfy the
functional equation (<ref>) for some β>0 then there is a
(unique) SGM μ^h_β,ξ.




 §.§ Uniqueness results

From now on, we confine ourselves to the case of uniform
(ferromagnetic) nearest-neighbour interaction by setting
J_xy=J≥ 0 if d(x,y)=1 (and J_xy=0 otherwise). It will also
be convenient to re-parameterize the model by introducing the new
parameter θ=^β J≥1 termed activity.

For θ≥ 1, consider the function

    φ(t;θ):=(θ-1) t/(√(θ
    (t-1))+√(t-θ))^2,     t≥θ,

which can also be written as

    φ(t;θ)= √(θ(t-1))-√(t-θ)/√(θ(t-1))+√(t-θ),     t≥θ.

Noting that

    φ(t;θ)=θ-1/(√(θ-θ/t)+√(1-θ/t))^2,

it is evident that t↦φ(t;θ) is a decreasing
function; in particular, for all t≥θ

    1=φ(θ;θ)≥φ(t;θ)≥φ(∞;θ)=√(θ)-1/√(θ)+1.


For brevity, introduce the notation

    Q(θ):=(q-2)√(θ)-1/√(θ)+1,

and for k≥2, q≥2 consider the equation

    Q(θ)+φ(θ+1;θ)=1/k,    θ≥ 1,

or more explicitly (noting that
φ(θ+1;θ)=(θ-1)/(θ+1)),

    Q(θ)+θ-1/θ+1=1/k,    θ≥ 1.

The left-hand side of (<ref>) is a continuous increasing
function of θ∈[1,∞) ranging from 0 to q-1>k^-1,
which implies that there is a unique solution of the equation
(<ref>), denoted θ_0=θ_0(k,q). In particular,
for q=2 we get

    θ_0(k,2)=k+1/k-1.


Let us also consider the equation

    (q-1)θ-1/θ+1=1/k,

which has the unique root

    θ_*=θ_*(k,q):=k(q-1)+1/k(q-1)-1.

Noting from (<ref>) that, for any θ>1,

    (q-1)√(θ)-1/√(θ)+1<
    Q(θ)+θ-1/θ+1≤
    (q-1)θ-1/θ+1,

and comparing equations (<ref>) and (<ref>), it
follows that

    θ_*(k,q)≤θ_0(k,q)< (θ_*(k,q))^2,

where the first inequality is in fact strict unless q=2.


Let θ_0=θ_0(k,q) be the  unique solution of the
equation (<ref>). Then the Gibbs measure
μ_θ,ξ is unique for θ∈(1,θ_0) and any
external field ξ.



It is known that the Ising model on a Cayley tree with zero external
field has a unique Gibbs measure if and only if θ≤θ_ c(k)=√(1+2/k-1)
 (see <cit.>); that is to say, θ_ c(k) is the
critical activity of the Ising model. Since
θ_ c(k)=θ_0(k,2), our Theorem <ref> is
sharp in this case. Let θ_ cr(k,q) be the critical
activity for the Potts model; its exact value is known only for the
binary tree (k=2), namely θ_
cr(2,q)=1+2√(q-1) <cit.>. Note that θ_
cr(2,2)=θ_0(2,2) (= 3) but θ_
cr(2,q)>θ_0(2,q) for q≥ 3, so Theorem <ref> is not
sharp already for k=2, q≥ 3.


For k≥ 2, q≥ 2 and any γ∈ℝ, consider the
equation (cf. (<ref>))

    Q(θ)+
    φ(t_γ(θ);θ)=1/k,    θ≥1,

where φ(t;θ) and Q(θ) are defined in
(<ref>) and (<ref>), respectively, and

    t_γ(θ):=θ+1+(q-2)θ^γ.

It can be shown (see Lemma <ref>) that equation
(<ref>) has a unique root,
θ^*_γ=θ^*_γ(k,q). More
specifically, if q=2 then t_γ(θ)=θ+1 and equation
(<ref>) is reduced to equation
(<ref>) (with Q(θ)≡ 0), so that
θ_γ^*(k,2)≡θ_0(k,2)=(k+1)/(k-1)
(see (<ref>)). However, if q≥ 3 then the root
θ^*_γ is an increasing function of parameter
γ with the asymptotic bounds

    θ_0(k,q)=lim_γ→-∞θ^*_γ(k,q)<θ^*_γ(k,q)<
    lim_γ→+∞θ^*_γ(k,q)=(θ_*(k,q))^2.



Given the external field
ξ(x)=(ξ_1(x),…,ξ_q(x)) (x∈ V), define
the asymptotic “gap” between its coordinates as follows,

    ^ξ:=max_1≤ℓ≤ qlim inf_x∈ V _ℓξ̌_(1)(x),

where

    _ℓξ̌_(1)(x):=min_iℓ_ℓξ̌_i(x)≡min_iℓ
    (ξ_i(x)-ξ_ℓ(x)),     x∈ V.



The Gibbs measure μ_β,ξ is unique for any
β∈(0,lnθ^*_^ξ-k), where
θ_γ^* denotes the unique  solution of the equation
(<ref>).




As already mentioned, if q=2 then
θ_γ^*(k,2)≡θ_0(k,2) and we recover Theorem
<ref> in this case. But if q≥ 3 then
θ_γ^*(k,q)>θ_0(k,q) for any
γ∈ℝ (see (<ref>)), so that
Theorem <ref> ensures the uniqueness of the SGM
μ_β,ξ on a wider interval of temperatures as
compared to Theorem <ref>, for any ^ξ. Moreover, due
to the monotonicity of the map
γ↦θ_γ^* (Lemma <ref>), a
larger gap ^ξ facilitates uniqueness of SGM; however,
the domain of guaranteed uniqueness (in parameter θ) is
bounded in all cases (see (<ref>)) by
(θ_*(k,q))^2≤ (θ_*(2,3))^2=25/9≐ 2.7778.


 If the external field ξ is random then
the gap (<ref>) is a random variable measurable with
respect to the “tail” σ-algebra
ℱ^∞=⋂_n=0^∞σ{ξ(x), x∈ V_n^c}. Intuitively, this means
that ^ξ does not depend on the values of the
field ξ(x) on any finite set ⊂ V.
If the values of ξ(x) are assumed to be independent
(not necessarily identically distributed) for different x∈ V
then, by Kolmogorov's zero–one law, ^ξ= (and
therefore θ^*_^ξ-k=) almost surely
(a.s.).



Let us compute the asymptotic gap ^ξ in a few examples.


  (a) Let the random vectors ξ(x) (x∈ V) be mutually
independent, with independent and identically distributed (i.i.d.)coordinates ξ_i(x) (i=1,…,q), each taking the values
±1 with probabilities 1/2. Note that
_ℓξ̌_(1)(x)∈{0,±2}  (ℓ=1,…,q) and

    (_ℓξ̌_(1)(x)=-2)   =
    (ξ_ℓ(x)=1, min_iℓξ_i(x)=-1)
       =(ξ_ℓ(x)=1)·(1-(ξ_i(x)=1, iℓ))
       =12(1-(12)^q-1)>0.

The Borel–Cantelli lemma then implies that lim inf_x∈
V_ℓξ̌_(1)(x)=-2 a.s. and hence, according to
(<ref>), ^ξ=-2 a.s.


  (b) In the previous example, let us remove the i.i.d.assumption for the coordinates, and instead suppose that each of the
random vectors ξ(x) (still mutually independent for
different x∈ V) can take two values, ±(1,…,1), with
probability 1/2 each. Then it is clear that
_ℓξ̌_(1)(x)≡0 (ℓ=1,…,q), hence
^ξ=0 a.s.


  (c) Extending example (b), suppose
that, with some α∈ℝ,

    (ξ_i(x)=α and ξ_j(x)=0 for allj i)=q^-1     (i=1,…,q).

Of course, for α=0 we have ξ(x)≡0 and hence ^ξ=0; thus, let α0.
If q=2 then it is straightforward to see that

    (_ℓξ̌_(1)(x)=±α)=12,    ℓ=1,2.

For q≥3, if α>0 then,
similarly,

    (_ℓξ̌_(1)(x)=-α)=q^-1,    (_ℓξ̌_(1)(x)=0)=1-q^-1,

whereas if α<0 then

    (_ℓξ̌_(1)(x)=-α)=q^-1,    (_ℓξ̌_(1)(x)=α)=1-q^-1.

Thus, in all cases, the Borel–Cantelli lemma yields that
^ξ=-|α| a.s. (which also includes the case
α=0).


  (d) Consider i.i.d. vectors ξ(x) (x∈ V) with i.i.d.coordinates ξ_i(x) (i=1,…,q), each with the uniform
distribution on [0,1]. Note that -1≤_ℓξ̌_(1)(x)≤ 1 (ℓ=1,…,q) and, for any
ε∈(0,1),

    (_ℓξ̌_(1)(x)≤ -1+2ε)   ≥(ξ_ℓ(x)≥
    1-ε, min_iℓξ_i(x)≤ε)
       =(ξ_ℓ(x)≥ 1-ε)·(1-
    {min_iℓξ_i(x)≥ε})
       =ε(1-(1-ε)^q-1)>0.

The Borel–Cantelli lemma then implies that lim inf_x∈
V_ℓξ̌_(1)(x)≤ -1+2ε a.s., and since
ε>0 is arbitrary, it follows that lim inf_x∈
V_ℓξ̌_(1)(x)=-1 a.s., for each ℓ=1,…,q;
hence, according to (<ref>), ^ξ=-1
a.s.






  (e) For a “non-ergodic” type of example leading to
a random gap ^ξ, suppose that
ξ(x)≡ξ(x_∘) (x∈ V),
where the distribution of ξ(x_∘) is as in
example (a). That is to say, the values of the field
ξ(x) are obtained by duplicating its (random) value
at the root. Then, similarly to (<ref>), we compute

    (^ξ=-2)= 12-(12)^q,    (^ξ=2)= (12)^q,    (^ξ=0)=12.



  (f) Finally, the simplest “coordinate-oriented” choice
ξ(x)≡(α,0,…,0), x∈ V, with a
fixed α∈ℝ, exemplifies translation-invariant
(non-random) external fields, including the case of zero field,
α=0. Our results for this model will be stated in
Section <ref>; for now, let us calculate the value of the
gap ^ξ. Again, for α=0 we have
ξ(x)≡0 and hence
^ξ=0; thus, let α0. If q=2 then
_1ξ̌_(1)(x)=-α,
_2ξ̌_(1)(x)=α, hence it is easy to see that
^ξ=max{-α,α}=|α|. For q≥ 3, if
α>0 then _1ξ̌_(1)(x)=-α and
_ℓξ̌_(1)(x)=0 (ℓ1), whereas if α<0
then still _1ξ̌_(1)(x)=-α but
_ℓξ̌_(1)(x)=α (ℓ1); as a result,
^ξ=0 for α≥0 and ^ξ=|α| for
α<0.



The following general assertion summarizes Example <ref>.
Recall that random variables X_1,…,X_q are said to be
exchangeable if the distribution of the random vector
(X_1,…,X_q) is invariant with respect to permutations of the
coordinates. The support X of (the distribution of) a
random variable X is defined as the (closed) set comprising all
points u∈ℝ such that for any ε>0 we have
ℙ(|X-u|≤ε)>0.

Suppose that the random vectors {ξ(x)}_x∈ V
are i.i.d., and for each x∈ V their coordinates
ξ_1(x),…,ξ_q(x) are exchangeable. Then

    ^ξ=inf{(ξ_1(x)-ξ_q(x))}  a.s.

In particular, ^ξ<0 a.s., unless
ξ_1(x)=…=ξ_q(x) a.s., in which case  ^ξ=0
a.s.


Observe that, by exchangeability of {ξ_i(x)}, the
distribution of _ℓξ_(1)(x) does not depend on
ℓ=1,…,q and, moreover,

    _ℓξ̌_(1)(x)=(ξ_1(x)-ξ_q(x)).

Denote u_0:=inf{(ξ_1(x)-ξ_q(x))}. From
(<ref>), it follows that _ℓξ̌_(1)(x)≥
u_0 a.s., and therefore, according to (<ref>),
^ξ≥ u_0 a.s. On the other hand, for any
ε>0 we have

    (_ℓξ̌_(1)(x)≤ u_0+ε)=
    (ξ_1(x)-ξ_q(x)≤ u_0+ε)>0,

and the Borel–Cantelli lemma implies that lim inf_x∈
V_ℓξ̌_(1)(x)≤ u_0+ε a.s., so that
^ξ≤ u_0 a.s. As a result, ^ξ=u_0 a.s.,
as claimed.



Assume that the random external field
ξ={ξ(x)}_x∈ V is as in
Proposition . Let
θ^†=θ^†(k,q) be the root of the
equation[The left-hand side of
(<ref>) does not depend on x∈ V due to the
i.i.d. assumption on {ξ(x)}.]

    Q(θ)+𝔼{φ(t_ξ̌_(1)(x)-k(θ);θ)}
    =1/k,    θ≥1,

where
ξ̌_(1)(x)≡_qξ̌_(1)(x)=min_i
q(ξ_i(x)-ξ_q(x))
cf. (<ref>) and the notation
t_γ(θ) is introduced in (<ref>). Then, for
each θ∈[1,θ^†) and for ℙ-almost all
realizations of the random field ξ, there is a
unique Gibbs measure μ_θ,ξ.



Note that Theorem <ref> guarantees uniqueness of the Gibbs
measure in the interval 1<θ<θ^*_^ξ-k,
where θ^*_^ξ-k is the solution of the
equation

    Q(θ)+φ(t_^ξ-k(θ);θ)=1/k,    θ≥1.

By Proposition <ref>, we have ξ̌_(1)(x)≥^ξ (a.s.), and moreover, ξ̌_(1)(x)>
^ξ with positive probability, unless
ξ_1(x)=…=ξ_q(x) a.s. Thus, excluding the case q=2 where
t_γ(θ)=θ+1, by monotonicity of the function
γ↦φ(t_γ;θ) we conclude that
θ^*_^ξ-k(k,q)<θ^†(k,q), and
therefore the domain of uniqueness in Theorem <ref> is wider than
that in Theorem <ref>.



Let us illustrate Theorem <ref> with a simple model described in
Example <ref>(c), assuming that q≥3. Suppose first that α>0.
Then, according to the distribution (<ref>) and notation (<ref>),
equation (<ref>) specializes to

    Q(θ)+1/q φ(θ+1+
    q-2/θ^α+k_;θ)+
    q-1/q φ(θ+1+q-2/θ^k;θ)
    =1/k.

By monotonicity of the function t↦φ(t;θ), it is
clear that the root θ^† of this equation is strictly
bigger than the root
θ^*_^ξ-k=θ^*_-α-k of the
equation

    Q(θ)+φ(θ+1+q-2/θ^α+k_;θ)
    =1/k,

in accordance with Remark <ref>. Similarly, if α<0
then the distribution (<ref>) is replaced by
(<ref>) and equation (<ref>) takes
the form

    Q(θ)+1/q φ(θ+1+
    q-2/θ^α+k_;θ)+
    q-1/q φ(θ+1+q-2/θ^-α+k_;θ)
    =1/k,

and by the monotonicity argument it is evident that its root
θ^† is strictly bigger than the root
θ^*_^ξ-k=θ^*_α-k of the
equation

    Q(θ)+φ(θ+1+q-2/θ^-α+k;θ)
    =1/k,

again confirming the observation of Remark <ref>.




 §.§ Translation-invariant SGM and the problem of
non-uniqueness




  §.§.§ Translation invariance

To introduce the notion of translations on the Cayley tree
𝕋^k, let 𝒜_k be the free group with
generators a_1, …, a_k+1 of order 2 each (i.e.,
a_i^-1=a_i). It is easy to see (cf. <cit.> and also
<cit.>) that the Cayley tree 𝕋^k=(V,E) is
in a one-to-one correspondence with the group 𝒜_k.
Namely, start by associating the root x_∘∈ V with the
identity element e∈𝒜_k, and identify the elements
a_1,…,a_k+1 with the k+1 nearest neighbours of x_∘
(i.e., comprising the set S(x_∘)=W_1). Proceed inductively by
expanding the elements a∈𝒜_k along the tree via
right-multiplication by the generators a_i
(i=1,…,k+1), yielding k new elements[Indeed, if
a=w a_j then a a_j=w a_j^2=w, so this particular
multiplication returns the element w already obtained at the
previous step.] corresponding to the set of direct successors of a
(see Figure <ref>). This establishes a bijection
𝔟 V→𝒜_k.

























Consider the family of left shifts T_g𝒜_k→𝒜_k (g∈𝒜_k) defined by

    T_g(a):=g a,     a∈𝒜_k.

By virtue of the bijection 𝔟, this determines conjugate
translations on V,

    T_z:= 𝔟^-1∘
    T_𝔟(z)∘𝔟,     z∈ V.

Clearly, T_z is an automorphism of V preserving the
nearest-neighbour relation; indeed, if ⟨ x,y⟩∈ E and
y∈ S(x) (so that 𝔟(y)=𝔟(x) a_j,
with some generator a_j) then, according to (<ref>),
x':=T_z(x) and y':=
T_z(y)=𝔟^-1(𝔟(z)𝔟(y))=𝔟^-1(𝔟(x') a_j) are
nearest neighbours, ⟨ x',y'⟩∈ E (see
Figure <ref>). For example, if k=1 (whereby the Cayley tree
𝕋^1 is reduced to the integer lattice ℤ^1),
the action of the shift T_z (z∈ℤ^1) can
be written in closed form,

    T_z(x)=z+(-1)^z x,     x∈ℤ^1.


In turn, the map (<ref>) induces shifts on
configurations σ∈^V,

    (T_zσ)(x):=σ(T^-1_zx),
         x∈ V.











We say that SGM μ^h=μ^h_θ,ξ is translation
invariant (with respect to the group of shifts
(T_z)) if for each z∈ V the measures μ^h∘T_z^-1 and μ^h coincide; that is, for any
(finite) ⊂ V and any configuration ς∈^

    μ^h(σ_T_z()=T_z(ς))
    =μ^h(σ_=ς).








Recall that the quantities ȟ_i^†(x,y)
⟨ x,y⟩∈ E defining the boundary
law were introduced in Section <ref>.

An SGM μ^h=μ^h_θ,ξ is translation invariant
under the group of shifts (T_z)_z∈ V if and only
if the following conditions are satisfied.


  (i) The reduced field {ξ̌(x)}
is constant over the tree,

    ξ̌(x)=
    ξ̌(x_∘),     x∈ V.


  (ii) The reduced field {ȟ^†(x,y)} is symmetric,

    ȟ^†(x,y)=
    ȟ^†(y,x),    ⟨ x,y⟩∈ E.



  (iii)
For any z∈ V,

    ȟ^†(x,y)=
    ȟ^†(T_z(x),T_z(y)),    ⟨ x,y⟩∈ E.




This result will be proved in Appendix <ref>.

For x x_∘, denote by x' the unique vertex such that
x∈ S(x'). Then, according to (<ref>)
and (<ref>),

    ȟ^†(x,x')=ȟ(x)
    =ȟ(T^-1_x'(x)).

Note that T^-1_x'(x)∈∂{x_∘}=W_1. Thus,
there are k+1 (vector) values ȟ(x_j)
(x_j∈ W_1) that determine a translation-invariant SGM μ^h.
By translations (<ref>), these values (which can be
pictured as k+1 “colours”) are propagated to all vertices x∈
V in a periodic “chessboard” tiling, except the root x=x_∘
where the value ȟ(x_∘) is calculated
separately, according to the compatibility formula (<ref>).

The criterion of translation invariance given by
Proposition <ref> appears to be new. In Georgii
<cit.>, 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 𝕋^k.[It
is worth pointing out that the latter group is generated by the
group of (left) shifts (T_z) and pairwise inversions
between vertices x_j,x_ℓ∈∂{x_∘} <cit.>.] Namely, we have the following


An SGM μ^h=μ^h_θ,ξ is completely homogeneous if
and only if ξ̌(x)≡ξ̌^0 for all x∈ V and
ȟ(x)≡ȟ^0
for all x x_∘.



In the existing studies of Gibbs measures on trees (see, e.g.,
<cit.>), it is common to use the term “translation
invariant” (and the abbreviation TISGM) having in mind just
completely homogeneous SGM. We prefer to keep the terminological
distinction between “single-coloured” completely homogeneous GBC
ȟ(x)≡ȟ^0
and “multi-coloured” translation-invariant GBC as characterized by
Proposition <ref>. 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.





  §.§.§ Analysis of uniqueness

For the rest of Section <ref>, we deal with completely
homogeneous SGM μ^h=μ^h_θ,ξ, that is, with the
external field {ξ(x)}_x∈ V and GBC
{h(x)}_x∈ V satisfying the homogeneity
conditions of Corollary <ref>. For simplicity, we
confine ourselves to the case where all coordinates of the (reduced)
vector ξ̌^0 are zero except one; due
to permutational symmetry, we may assume, without loss of
generality, that ξ̌_1^0=α∈ℝ and
ξ̌^0_2=…=ξ̌^0_q-1=0,

    ξ̌^0 = (α,0,…,0)∈ℝ^q-1.

We also write

    ȟ^0=(ȟ_1^0,…,ȟ_q-1^0)∈ℝ^q-1.

Then, denoting z_i:=θ^ȟ_i^0/k
(i=1,…,q-1), the compatibility equations (<ref>) are
equivalently rewritten in the form

    { z_1   =1+(θ-1)(θ^α
    z_1^k-1)/θ+θ^α_
    z_1^k+∑_j=2^q-1z_j^k,
    
    z_i   =1+(θ-1)(z_i^k-1)/θ+θ^α_
    z_1^k+∑_j=2^q-1z_j^k,     i=2,…,q-1.
    .


Solvability of the system (<ref>) 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 θ=1 is trivial, as the system (<ref>) will then
have the unique solution z_1=…=z_q-1=1. The case θ>1
and α=0 has been studied in <cit.>; these results can be
reproduced directly by the methods developed in the present work
similarly to a more general (and difficult) case α0, and
are also obtainable in the limit as α→0 (see
Lemma <ref>(b) and Remark <ref>).

Thus, let us focus on the new case α 0. By
Lemma <ref>(c), the system (<ref>) with θ>1 is
reduced either to a single equation

    u=1+(θ-1)(θ^α
    u^k-1)/θ+θ^α_
    u^k+q-2

or to the system of equations (indexed by m=1,…, q-2)

    { u   =1+(θ-1)(θ^α
    u^k-1)/θ+θ^α_
    u^k+m v^k+q-2-m,
    
    1   =(θ-1)(1+v+…+v^k-1)/θ+θ^α_
    u^k+m v^k+q-2-m,
    .

subject to the condition

    v1.


Let us first address the solvability of the
equation (<ref>). Denote

    θ_ c=θ_ c(k,q):=
    1/2 (√((q-2)^2+4(q-1)(k+1/k-1)^2)-(q-2)).

In particular,  if q=2 then θ_
c(k,2)=k+1/k-1≡θ_0(k,2)
(cf. (<ref>)). Let us also set

    b= b(θ):=θ(θ+q-2)/q-1.

Clearly, b(1)=1 and b(θ)>1 for θ>1. Furthermore,
comparing (<ref>) and (<ref>) observe that
b(θ_ c)=(k+1/k-1)^2 and
b(θ)>(k+1/k-1)^2 for θ>θ_
c. For θ≥θ_ c, denote by x_±=
x_±(θ) the roots of the quadratic equation

    (b+x)(1+x)=k(b-1) x

with discriminant

    D= D(θ):=(k(b-1)-(b+1))^2-4b=
    (b-1)(k-1)^2(b-(k+1/k-1)^2),

that is,[Here 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.]

    x_±= x_±(θ):=(b-1)(k-1)-2±√(D)/2.

Furthermore, introduce the notation

    a_±=
    a_±(θ):=1/x_±(1+x_±/b+x_±)^k,    θ≥θ_ c.

Of course, a_-(θ_ c)=a_+(θ_ c), and one
can also show that a_-(θ)<a_+(θ) for all
θ>θ_ c (see the proof of Theorem <ref> in
Section <ref>). Finally, denote

    α_±=α_±(θ):=-(k+1)+1/lnθlnq-1/a_∓,    θ≥θ_ c,

so that α_-(θ_ c)=α_+(θ_ c) and
α_-(θ)<α_+(θ) for θ>θ_ c.


Let ν_0(θ,α) denote the number of solutions u>0 of
the equation (<ref>). Then

    ν_0(θ,α)= 1   if  θ≤θ_ c  or  θ>θ_ c  and  α∉ [α_-,α_+],
    
    2   if  θ>θ_ c  and  α∈{α_-,α_+},
    
    3   if  θ>θ_ c  and  α∈
    (α_-,α_+),

where θ_ c is given in (<ref>) and
α_±=α_±(θ) are defined by (<ref>).


Let us now state our results on the solvability of the set of
equations (<ref>). For each m∈{1,…,q-2}, consider
the functions

    L_m(v;θ):=   (θ-1)(v^k-1+…+v)-m v^k
    -(q-1-m),
     K_m(v;θ):=   (v^k-1+…+v+1)^k
    L_m(v;θ)/(v^k-1+…+v+L_m(v;θ))^k.

It can be checked (see Lemma <ref>) that for any
θ>1 there is a unique value v_m= v_m(θ)>0 such that

    L_m^*(θ):=L_m(v_m;θ)=max_v>0L_m(v;θ),

and moreover, the function θ↦ L_m^*(θ) is strictly
increasing. Denote by θ_m the (unique) value of θ>1
such that

    L_m^*(θ_m)=0.

Thus, for any θ>θ_m the range of the functions v↦
L_m(v;θ) and v↦ K_m(v;θ) includes positive
values,

    𝒱_m^+(θ):={v>0 L_m(v;θ)>0}≡{v>0 K_m(v;θ)>0}∅,

and, therefore, the function

    α_m(θ):=1/lnθmax_v∈𝒱_m^+(θ)ln K_m(v;θ)=ln
    K_m^*(θ)/lnθ,    θ>θ_m,

is well defined, where

    K_m^*(θ):=max_v∈𝒱_m^+(θ) K_m(v;θ).



In the case k=2, from (<ref>) and (<ref>) we
obtain explicitly

    θ_m=1+2√(m(q-m-1)),      m=1,…,q-2.

In particular, θ_1=1 for q=2, and θ_1≥ 3 whenever
q≥3. Comparing (<ref>) and (<ref>), we
also find that θ_ c≤θ_1 if and only if
q≥ 6. For example, for q=5 we have θ_1=1+2√(3)≐
4.4641, θ_2=5, θ_
c=1/2(√(153)-3)≐ 4.6847, that is,
θ_1<θ_ c<θ_2, whereas for q=6 and
q=7 we compute θ_1=θ_ c=5 and
θ_1=1+2√(5)≐5.4721>θ_
c=1/2(√(241)-5)≐5.2621, respectively. Another
simple case is q=3 (with m=1 and any k≥2); indeed, it is
easy to see that the condition (<ref>) is satisfied with
v_1^*=v_1(θ_1)=1 (cf. Lemma <ref>(c)), whence we
readily find θ_1=1+2/k-1.


 For each m∈{1,…,q-2},
let ν_m(θ,α) denote the number of positive
solutions (u,v) of the system (<ref>). Then
ν_m(θ,α)≥ 1 if and only if θ>θ_m and
α≤α_m(θ).


As will be shown in Proposition <ref>(b),
α_+(θ) has a unique zero given by
θ^+_0=1+q/k-1, whereas
α_-(θ) has a unique zero θ^-_0, which
coincides with the zero θ_1^0
of α_1(θ). It also holds that α_1(θ)
is a majorant of the family of functions {α_m(θ)} (see
Propositions <ref> and <ref>).




  §.§.§ Uniqueness of completely homogeneous SGM

In the case q=3, there appears to be an additional
critical value (see Lemma <ref>)

    θ̃_1=θ̃_1(k):=5-k+√(49k^2+62k+49)/6(k-1).

For example,

    θ̃_1(k)=1+√(41)/2≐ 3.7016,    k=2,
     7/3≐ 2.3333,   k=3,
    1+√(1081)/18≐ 1.8821,     k=4.


For q≥2, consider the following subsets of the half-plane
{θ≥1}={(θ,α)θ≥1, α∈ℝ},

    A_q:=   {θ>θ_ c, α_-(θ)≤α≤α_+(θ)},
     B_q:=   {3    ∅      if  q=2,
       {θ>θ_1, α<α_1(θ)}∪{θ>θ̃_1,
     α=α_1(θ)}        if  q=3,
       {θ>θ_1, α≤α_1(θ)}        if q≥4.
    .

Denote the total number of positive solutions
z=(z_1,…,z_q-1) of the system of
equations (<ref>) by ν(θ,α) (θ≥1,
α∈ℝ); of course, this number also depends on k
and q. Theorems <ref> and <ref> can now be
summarized as follows.




  (a)
If q=2 then ν(θ,α)≥2 if and only if
(θ,α)∈ A_2∪ B_2=A_2.




  (b) If q=3 then
ν(θ,α)≥ 2 if (θ,α)∈ A_3∪ B_3. The“only if” statement holds true at least for k=2,3,4.




  (c) If q≥ 4 then
ν(θ,α)≥ 2 if and only if (θ,α)∈ A_q∪
B_q.




The special case q=3 in the definition (<ref>) and in
Theorem <ref> emerges because for θ≤θ̃_1 and α=α_1(θ), there is a
(hypothetically unique) solution
(u,v)=(θ-θ+2/k,1) of the
system (<ref>), which is, however, not admissible due to the
constraint (<ref>) and, therefore, does not
destroy the uniqueness of solution to (<ref>). This hypothesis
is conjectured below; if it is true then “if” in
Theorem <ref>(b) (i.e., q=3) can be enhanced to “if and
only if” for all k≥2.



If q=3, θ≤θ̃_1 and α=α_1(θ) then
(u,v)=(θ-θ+2/k,1) is the sole solution
of the system (<ref>).



Regardless of Conjecture <ref>, the inclusion of a proper
part of the curve α=α_1(θ) in the uniqueness region
in the case q=3 is indeed necessary. Namely, it can be proved (see
Proposition <ref>) that if ε>0 is small
enough then ν(θ,α_1(θ))=1 for θ_1<θ<
θ_1+ε but ν(θ,α_1(θ))≥ 2 for
θ>θ^0_1-ε, where θ_1^0 is the zero of
α_1(θ).


Theorem <ref> provides a sufficient and (almost) necessary
condition for the uniqueness of solution of (<ref>), illustrated
in Figure <ref> for q=5 and q=3, both with k=2.



To conclude this subsection, the following result describes a few
cases where it is possible to estimate the maximal number of
solutions of the system (<ref>).



  (a)
If q=2 then ν(θ,α)≤3 moreover,
ν(θ,α)=3 for all θ≥1 large enough.


  (b) Let α=0 and k≥2. Then ν(θ,0)≤ 2^q-1 for all
θ≥1 moreover, ν(θ,0)=2^q-1 for all
θ large enough.


  (c) If k=2 then ν(θ,α)≤ 2^q-1 for all
θ≥1 and α∈ℝ.




The upper bound 2^q-1 in Theorem <ref> appears to be
universal. There is empirical evidence from exploration of many
specific cases (using the computing package[Throughout this
paper, we used Maple 18 (Build ID 922027) licensed to the
University of Leeds.] Maple) to conjecture that
Theorem  holds true for all k≥2.




  §.§.§ Some comments on earlier work


For q=2, when the Potts model is reduced to the Ising model, the
result of Theorem <ref>(a) is well known
(see <cit.> or <cit.>).

The case α=0 (i.e., with zero field) is also well studied
(see, e.g., <cit.> and <cit.>), the result of
Theorem <ref>(b) can be considered as a corollary of
<cit.>); in particular, it is known that
there are ⌊1/2(q+1)⌋ critical values of
θ, including θ_0^-=θ_1^0 and
θ_0^+=1+q/(k-1).

The general case q≥3 with α∈ℝ was first
addressed by Peruggi et al. <cit.> (and continued
in <cit.>) using physical argumentation. In particular,
they correctly identified the critical point θ_ c
<cit.> (cf. (<ref>))
and also suggested an explicit critical boundary in the phase
diagram for α≥0, defined by the expression
<cit.> (adapted to our notation)

    α̃_-(θ)=(k+1)ln(1+(q-2)/θ)-(k-1)ln(q-1)/2lnθ.

Note that this function enjoys a correct value at
θ=θ_ c (i.e., α̃_-(θ_
c)=α_±(θ_ c), see formula (<ref>)
below), but α̃_-(θ)>α_-(θ) for all
θ>θ_ c. The corresponding critical value of
activity θ, emerging as zero of α̃_-(θ),
is reported in <cit.> as

    θ_ cr=θ_
    cr(k,q):=q-2/(q-1)^(k-1)/(k+1)-1.

In particular, θ_ cr is bigger than the exact critical
value θ_0^-=θ_1^0, where the uniqueness breaks
down at α=0 (see Proposition <ref>). For
example, the corresponding numerical values (for k=5 and q=3 or
q=8) are given by (cf. <cit.>)

    θ_ cr≐
    1.7024
    
    ,   q=3,
    
    2.2562
    
    ,   q=8,
        θ_0^-≐1.6966
    
    ,   q=3,
    
    2.1803
    
    ,   q=8.


The critical boundary in the phase diagram for α≤0 was
described in <cit.> only heuristically, as a line
“joining” the points θ=θ_ cr(k,q), α=0 and
θ=θ_ cr(k,q-1), α=-∞, and illustrated
by a sketch graph in the vicinity of θ_ cr(k,q) (for
k=5 and q=3 or q=8).

It should be stressed that the phase transition occurring at these
critical boundaries is not of type “uniqueness/non-uniqueness”,
with which we are primarily concerned in the present paper, but in
fact the so-called “order/disorder” phase transition. The latter
was studied rigorously in a recent paper by Galanis et al.<cit.> in connection with the computational complexity of
approximating the partition function of the Potts model. The useful
classification of critical points deployed in <cit.> is
based on the notion of dominant phase; in particular, the
critical point θ_0^+=1+q/(k-1) (conjectured earlier by
Häggström <cit.> 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 <cit.> studies the Potts model primarily with
zero external field (α=0); 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 θ_ cr 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 of <cit.>. More specifically, our
analysis (see Proposition <ref>) shows that the
critical point θ_0^+ is the signature of the upper critical
function α_+(θ) at α=0, which has a minimum at
θ=θ_0^+. Therefore, it is reasonable to
conjecture that the α>0 analogue of the interval of
activities θ between the critical points θ_0^- and
θ_ cr is the interval [θ_α^-,
θ_α^+], where θ_α^- is the (sole) root of
the equation α_-(θ)=α and θ_α^+ is
the smaller root of the similar equation
α_+(θ)=α. 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 α<0 is likely
to be simpler, as only the equation α_-(θ)=α is
involved. We intend to address these issues in our forthcoming work.



§ 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.


For a,b,c,d>0, consider the function

    f(t):=lna^t+b/c^t+d,     t∈ℝ.



  (a) If ad>bc then f(t) is monotone increasing on ℝ and

    lnb/d≤ f(t)≤lna/c,     t∈ℝ.

Similarly, if  ad<bc then f(t) is monotone decreasing on
ℝ and

    lna/c≤ f(t)≤lnb/d,     t∈ℝ.



  (b) Furthermore,

    |f'(t)|≤|ad-bc|/(√(ad)+√(bc))^2,     t∈ℝ.



(a) Differentiating equation (<ref>), we get

    f'(t)=a^t/a^t+b-c^t/c^t+d=ad-bc/ac^t+bd^-t+ad+bc.


If ad>bc then, according to (<ref>), the function f(t) is
monotone increasing, and the bounds (<ref>) follow by taking
the limit as t→±∞. The case ad<bc is similar.

(b) By the inequality between the arithmetic and geometric means,
the denominator on the right-hand side of (<ref>) is bounded
below by

    2√(abcd)+ad+bc=(√(ad)+√(bc))^2,

and the result (<ref>) follows.


Let us define two norms for vector
u=(u_1,…,u_q-1)∈ℝ^q-1,

    u_∞:=max_1≤ i≤ q-1 |u_i|,    u_1:=∑_i=1^q-1 |u_i|.

The next two lemmas give useful estimates for the function
F=(F_1,…,F_q-1) defined in (<ref>) and
for its partial derivatives.

For any θ≥1, the following uniform estimate holds,

    sup_u∈ℝ^q-1F(u;θ)_∞≤lnθ.



Note that F_i, as a function of u_i, may be represented
by formula (<ref>) with the coefficients

    a=θ,   b=1+s,   c=1,   d=θ+s,   s:=∑_j
    i^u_j≥0,

where

    ad-bc=θ(θ+s)-(1+s)=(θ-1)(θ+1+s)>0.

Therefore, by the estimates (<ref>) we have

    ln1+s/θ+s≤ F_i(u;θ)≤lnθ.

Furthermore, noting that (1+s)/(θ+s)≥ 1/θ, the
two-sided bound (<ref>) implies the inequality
|F_i(u;θ)|≤lnθ, and the bound (<ref>) follows by
taking the maximum over i=1,…,q.


Recall that the function φ(t;θ) is defined by
(<ref>). Denote by ∇ F _i the gradient of the map
u↦ F_i(u;θ),

    ∇ F_i(u;θ):=(∂
    F_i(u;θ)/∂ u_1,…, ∂
    F_i(u;θ)/∂ u_q-1),    u∈ℝ^q-1.

Recall that the norm ·_1 is defined in
(<ref>).

For γ∈ℝ and any
u=(u_1,…,u_q-1)∈ℝ^q-1 such that
min_1≤ i≤ q-1u_i≥γlnθ, it holds

    max_1≤ i≤ q-1∇ F_i(u;θ)_1≤
    Q(θ)+φ(t_γ(θ);θ),

where Q(θ) is defined in (<ref>) and
t_γ(θ)=θ+1+(q-2)θ^γ see
(<ref>). Moreover, the following uniform
estimate holds,

    max_1≤ i≤ q-1sup_u∈ℝ^q-1∇ F_i(u;θ)_1≤ Q(θ)+
    θ-1/θ+1.



Like in the proof of Lemma <ref>, let us represent
F_i(u;θ) by formula (<ref>) with the
coefficients (<ref>). Then by Lemma <ref>

    |∂ F_i(u;θ)/∂
    u_i|≤(θ-1)(θ+1+s)/(√(θ(θ+s))+√(1+s))^2=
    φ(θ+1+s;θ),

where s=∑_j i^u_j≥
(q-2)θ^γ=t_γ(θ)-θ-1. Hence, by
monotonicity of the map t↦φ(t;θ), from
(<ref>) it follows that

    |∂ F_i(u;θ)/∂
    u_i|≤φ(t_γ(θ);θ).


On the other hand, expressing F_i(u;θ) by
formula (<ref>) with

    a=1,   b=θ^u_i+1+s',   c=1,  
    d=θ+^u_i+s',   s':=∑_ℓ i,j^u_ℓ≥0,

by Lemma <ref> we obtain

    |∂
    F_i(u;θ)/∂ u_j|   ≤(θ-1)|^u_i-1|/(√(θ+^u_i+s')+√(θ^u_i+1+s'))^2.

If u_i>0 then the estimate (<ref>) specializes to

    |∂ F_i(u;θ)/∂
    u_j|   ≤(θ-1)(^u_i-1)/(√(θ+^u_i+s')+
    √(θ^u_i+1+s'))^2
       ≤θ-1/(√(1+(θ+1)/(^u_i-1))
    +√(θ+(θ+1)/(^u_i-1)))^2
       ≤θ-1/(√(1)+√(θ))^2=√(θ)-1/√(θ)+1.

Similarly, if u_i≤ 0 then 1≥^u_i>0 and from
(<ref>) we obtain

    |∂ F_i(u;θ)/∂
    u_j|   ≤(θ-1)
    (1-^u_i)/(√(θ+^u_i+s')
    +√(θ^u_i+1+s'))^2
       ≤θ-1/(√(θ)+√(1))^2=√(θ)-1/√(θ)+1.

Thus, combining (<ref>) and (<ref>), we have

    |∂ F_i(u;θ)/∂
    u_j|≤√(θ)-1/√(θ)+1,     i
    j.

As a result, according to the definition (<ref>) of the norm
·_1, the bounds (<ref>) and (<ref>)
imply the estimate (<ref>).

Finally, since

    lim_γ→-∞φ(t_γ(θ);θ)=φ(θ+1;θ)=θ-1/θ+1,

from (<ref>) we obtain (<ref>), and the proof of
Lemma <ref> is complete.



 Estimates similar to (<ref>)
were proved in <cit.>.















 For integer q≥2 and any
γ∈ℝ, the map [1,∞)∋θ↦φ(t_γ(θ);θ) is a continuous, strictly
increasing function with the range [0,1).

Denoting q̃:=q-2∈ℕ_0, by formula
(<ref>) we have

    φ(t_γ(θ);θ)=φ(θ+1+q̃θ^γ;θ)   =
    √(θ(θ+q̃θ^γ))-√(1+q̃θ^γ)/√(θ(θ+q̃θ^γ))+√(1+q̃θ^γ)
       =1-2(1+√(θ^2+q̃θ^γ+1/1+q̃θ^γ) )^-1.

Clearly, the function (<ref>) is continuous, so we only
need to show that

    A(θ):=θ^2+q̃θ^γ+1/1+q̃θ^γ,    θ≥ 1,

is an increasing function.

First of all, if q̃=0 then (<ref>) is reduced to
A(θ)=θ^2, so there is nothing to prove. Suppose that
q̃≥ 1. Differentiating (<ref>), it is easy to see
that

    A'(θ)=2θ+q̃θ^γ[θ+2+q̃θ^γ+(1-γ)(θ-1)]/(1+q̃θ^γ)^2,

and it is evident that the right-hand side of (<ref>) is
positive for all θ≥1 as long as γ≤1. On the other
hand, for γ≥1 by Bernoulli's inequality we have

    q̃θ^γ≥θ^γ=(1+(θ-1))^γ≥ 1+γ
    (θ-1),

and the expression in square brackets in (<ref>) is estimated
from below by (θ+2)+1+(θ-1)=2θ+2>0. Thus, in
all cases A'(θ)>0 for θ≥1, as required.

Finally, from (<ref>) we see that A(1)=1 and
lim_θ→∞A(θ)=∞, and it follows that the
range of the function (<ref>) is [0,1), which completes
the proof of the lemma.



For k≥ 2, q≥ 2 and any γ∈ℝ, the
equation (<ref>) has a unique root
θ^*_γ=θ^*_γ(k,q). If q=2 then
θ^*_γ(k,2)≡θ_0(k,2)=(k+1)/(k-1), where
θ_0(k,q) is the root of the equation (<ref>). For
q≥ 3, the root θ^*_γ is a continuous monotone
increasing function of γ, such that

    lim_γ→-∞θ^*_γ(k,q)=θ_0(k,q),    lim_γ→+∞θ^*_γ(k,q)=(θ_*(k,q))^2,

where θ_*(k,q) is the root of the equation
(<ref>) in particular,

    θ_0(k,q)<θ^*_γ(k,q)<(θ_*(k,q))^2.


The case q=2 is straightforward, so assume that q≥3.
Due to continuity and monotonicity of the function Q(θ) (see
(<ref>)) and by virtue of Lemma <ref>, the left-hand
side of equation (<ref>) is a continuous
increasing function of θ∈[1,∞), with the range
[0,q-1) because

    lim_θ↓ 1(
    Q(θ)+φ(t_γ(θ);θ))=0,    lim_θ→∞(Q(θ)+φ(t_γ(θ);θ))=q-1.

Hence, the equation (<ref>) always has a unique
solution, θ^*_γ=θ^*_γ(k,q). Since
t_γ(θ) is a continuous increasing function of γ,
while the map t↦φ(t;θ) is continuous and
decreasing, it follows that the root θ^*_γ is
continuous and increasing in γ.

Finally, observing that (see (<ref>)
and (<ref>))

    lim_γ→-∞φ(t_γ(θ);θ)=φ(θ+1;θ)=θ-1/θ+1,    lim_γ→∞φ(t_γ(θ);θ)=φ(∞;θ)=√(θ)-1/√(θ)+1,

and comparing equation (<ref>) with the limiting
equations as γ→±∞ (which have the roots θ_0
and θ_*^2, respectively), we obtain the required limits
(<ref>), and hence the asymptotic bounds
(<ref>) for θ^*_γ.




§ PROOFS OF THE MAIN RESULTS RELATED TO UNIQUENESS




 §.§ Proof of Theorem  (criterion of compatibility)


For shorthand, denote temporarily
ζ(x):=h(x)+ξ(x).
Suppose that the compatibility condition (<ref>) holds. On
substituting (<ref>),

it is easy to see that (<ref>) simplifies to

    ∏_x∈ W_n∏_y∈ S(x)∑_ω(y)∈exp{β(J_xyδ_σ_n(x),ω(y)+ζ_ω(y)(y))}=Z_n+1/Z_n∏_x∈ W_nexp{β h_σ_n(x)(x)},

for any σ_n∈^V_n. Consider the equality
(<ref>) on configurations
σ_n^1,σ_n^2∈^V_n that coincide everywhere in
V_n except at vertex x∈ W_n, where σ_n^1(x)=i≤ q-1
and σ_n^2(x)=q. Taking the log-ratio of the two resulting
relations, we obtain

    ∑_y∈
    S(x)lnexp{β(J_xy+ζ_i(y))}+∑_j
    iexp{βζ_j(y)}/exp{β(J_xy+ζ_q(y))}+∑_j=1^q-1exp{βζ_j(y)}=
    β(h_i(x)-h_q(x)),

which is readily reduced to (<ref>) in view of the notation
(<ref>) and (<ref>).

Conversely, again using (<ref>) and (<ref>), equation
(<ref>) can be rewritten in the coordinate form as follows,

    ∏_y∈ S(x)∑_j=1^qexp{β(J_xyδ_ij+ζ_j(y))}= a(x)exp{β
    h_i(x)},     i=1,…,q-1,

where (omitting the immaterial dependence on β,
h and ξ) we denote

    a(x):= exp{β h_q(x)}∏_y∈ S(x)∑_j=1^qexp{β(J_xyδ_qj+ζ_j(y))},     x∈ V.

Hence, using (<ref>) and setting A_n:=∏_x ∈ W_n
a(x), we get

    ∑_ω∈^W_n+1μ^h_n+1(σ_n∨ω)   =exp{-β H_n(σ_n)}/Z_n+1∏_x∈ W_n∏_y∈
    S(x)∑_j=1^qexp{β(J_xyδ_σ_n(x), j+ζ_j(y))}
       = A_n/Z_n+1exp{-β
    H_n(σ_n)+β∑_x∈
    W_nh_σ_n(x)(x)}=A_nZ_n/Z_n+1μ_n^h(σ_n).

Finally, observe that

    ∑_σ_n∈^V_n∑_ω∈^W_n+1μ_n+1^h(σ_n∨ω)
    =∑_σ_n+1∈^V_n+1μ_n+1^h(σ_n+1)=1,

whereas from the right-hand side of (<ref>) the same sum is
given by

    ∑_σ_n∈^V_nA_nZ_n/Z_n+1μ_n^h(σ_n)=A_nZ_n/Z_n+1.

Hence, A_nZ_n/Z_n+1=1 and formula (<ref>) yields
(<ref>), as required. This completes the proof of
Theorem <ref>.



 §.§ Preparatory results for the uniqueness of SGM

First, let us rewrite the functional equation (<ref>) in a form
more convenient for iterations. Recall that we assume J_xy≡
J>0 (d(x,y)=1) and use the notation θ=^β J.

Via the substitutions

    g(x)
    =F(βȟ(x)+βξ̌(x);θ)∈ℝ^q-1,
         x∈ V,

and

    ȟ(x)=β^-1∑_y∈
    S(x)g(y),     x∈ V,

equation (<ref>) is equivalent to the fixed-point equation

    g(x)=g(x),     x∈
    V,

where the mapping (ℝ^q-1)^V→(ℝ^q-1)^V is defined by

    g(x):=F(βξ̌(x)+∑_y∈
    S(x)g(y);θ),     x∈ V.


By means of (<ref>), the recursive equation (<ref>)
for ȟ can be written as (<ref>).
Substituting this into (<ref>) and using the notation
(<ref>), we see that g solves the functional
equation (<ref>). Conversely, if g satisfies
the equation (<ref>) then for ȟ defined
by (<ref>) we have, using (<ref>),

    βȟ(x)=∑_y∈
    S(x)g(y)   =∑_y∈ S(x)g(y)
       =∑_y∈
    S(x)F(βξ̌(y)+∑_z∈
    S(y)g(z);θ)
       =∑_y∈
    S(x)F(βξ̌(y)+βȟ(y);θ),

so that ȟ solves the equation (<ref>).
Thus, Lemma <ref> is proved.
In particular, Lemma <ref> implies that for the proof of
uniqueness of SGM it suffices to show that the equation
(<ref>) has a unique solution g(x) (x∈ V).

Let us state and prove one general result in the contraction case.
On the vector space  (ℝ^q-1)^V of
ℝ^q-1-valued functions on the vertex set V of the
Cayley tree 𝕋^k, introduce the sup-norm

    g_V:=sup_x∈ Vg(x)_∞=
    sup_x∈ Vmax_1≤ i≤ q-1|g_i(x)|,    g(x)=(g_1(x),…,g_q-1(x)).

Sometimes, we need the similar norm for functions restricted to
subsets ⊆ V,

    g_:=sup_x∈g(x)_∞,    g∈
    (ℝ^q-1)^.


The next lemma and its proof are an adaptation of a standard result
for ℓ^∞(ℝ).


For any subset ⊆ V, the space
(ℝ^q-1)^ is complete with respect to the
sup-norm (<ref>).

Let {g^n} be a Cauchy sequence in
(ℝ^q-1)^, that is, for any
ε>0 there is N∈ℕ such that for any n,m≥
N we have
g^n-g^m_<ε. In
particular, {g^n} is bounded,
g^n_≤ M<∞ for some M>0 and
all n∈ℕ. Note that every coordinate sequence
{g^n_i(x)} (i=1,…,q-1, x∈) is also a Cauchy
sequence (in ℝ) because, according to (<ref>),
|g^n_i(x)-g^m_i(x)|≤g^n-g^m_<ε;
hence, it converges to a limit which we denote g_i(x). Clearly,
|g_i(x)|≤ M and g_=sup_x∈max_i |g_i(x)|≤ M<∞.

Now, passing to the limit as m→∞ in each inequality
|g^n_i(x)-g^m_i(x)|<ε, we obtain |g^n_i(x)-g_i(x)|≤ε, which implies that
g^n-g_≤ε,
for all n≥ N. Since ε>0 is arbitrary, we conclude
that g^n-g_→0 as
n→∞, and the lemma is proved.


We also require the following simple estimate.

 Let f(u)ℝ^q-1→ℝ be a
C^1-function and ∇ f(u)=(∂
f(u)/∂ u_1,…,∂
f(u)/∂ u_q-1) its gradient. Then, for
any v,w∈ℝ^q-1,

    |f(w)-f(v)|≤w-v_∞ sup_u∈ℝ^q-1∇ f(u)_1.


Define the function
ψ(t):=f(v+t(w-v)),
t∈[0,1], then

    f(w)-f(v)=ψ(1)-ψ(0)=∫_0^1
    ψ'(t) t=∫_0^1∑_i=1^q-1∂
    f(v+t(w-v))/∂
    u_i(w_i-v_i) t,

whence the estimate (<ref>) readily follows.



Suppose that, for some θ>1,

    λ(θ):=kmax _1≤ i≤ q-1sup _u∈ℝ^q-1∇ F_i(u;θ)_1<1.

Then, for every realization of the field
ξ={ξ(x)}_x∈ V, the equation
(<ref>) has a unique solution.

Consider a mapping
=(_1,…,_q-1) of the space
(ℝ^q-1)^V to itself defined by formula (<ref>).
Solving the functional equation (<ref>) is then equivalent to
finding a fixed point of , that is,
g^*=g^*. The
lemma's hypothesis implies that  is a
contraction on (ℝ^q-1)^V_0^c; indeed, for any
functions g,g̅∈
(ℝ^q-1)^V and each i=1,…,q-1, we obtain, using
(<ref>) and Lemma <ref>,

    |_ig(x)-_ig̅(x)|   ≤sup_u∈ℝ^q-1∇
    F_i(u;θ)_1∑_y∈
    S(x)g(y)-g̅(y)_∞.

Noting that for x x_∘ the set S(x) contains exactly k
vertices, and recalling condition (<ref>) with
λ(θ)∈[0,1), it follows that

    g-g̅_V_0^c   =sup_x∈
    V_0^cg(x)-g̅(x)_∞
       ≤max_1≤ i≤ q-1sup_u∈ℝ^q-1∇
    F_i(u;θ)_1·
    kg-g̅_V_0^c=λ(θ) g-g̅_V_0^c.

Thus, because (ℝ^q-1)^V_0^c is a Banach space
(Lemma <ref>), the well-known Banach contraction
principle (e.g., <cit.>) implies that
g-g̅_V_0^c=0, that is,
g(x)=g̅(x) for all x∈ V_0^c. It
remains to notice that the value of the solution g(x)
at x=x_∘ is uniquely determined from formulas (<ref>)
and (<ref>) using the (unique) values outside
V_0={x_∘}. This completes the proof of Theorem <ref>.



The unique solution g^* can be obtained by iterations
<cit.>; for example, put g^0≡0 and define
g^n:=g^n-1
(n∈ℕ), then g^n→g^* as
n→∞ (i.e.,
lim_n→∞g^n-g^*_V=0).



It is straightforward to generalize Theorem <ref> to the case
where the vector βξ̌(x)+∑_y∈
S(x)g(y) (see (<ref>)) is guaranteed to be in
a convex domain B(x)⊆ℝ^q-1 for any function
g V→ℝ^q-1 from a suitable subspace
𝒟⊆ (ℝ^q-1)^V, such that 𝒟
is closed with respect to the norm ·_V and
(𝒟)⊆𝒟. In
that case, the supremum in (<ref>) should be taken over all
u∈ B(x),

    λ(θ):=ksup_x∈ Vmax _1≤ i≤
    q-1sup_u∈ B(x)∇
    F_i(u;θ)_1<1,

and the unique solution g^* automatically
belongs to 𝒟. For our purposes, it will
suffice to consider the balls
B(x)={u∈ℝ^q-1u-βξ̌(x)_∞≤
klnθ} and the corresponding subspace
𝒟={g∈(ℝ^q-1)^V g_V≤lnθ} (see Lemma <ref>).




 §.§ Proofs of Theorems <ref>, <ref> and <ref> (uniqueness)


By virtue of Remark <ref>, for the uniqueness in the class
of all Gibbs measure it suffices to prove it for SGMs.



  §.§.§ Proof of Theorem 

By virtue of the uniform bound (<ref>) of Lemma <ref>, for
every θ∈[1,θ_0) we have

    λ(θ)=kmax_1≤ i≤ q-1sup_u∈ℝ^q-1∇
    F_i(u;θ)_1≤
    k(Q(θ)+θ-1/θ+1)<1,

and the required result follows by Theorem <ref>.



  §.§.§ Proof of Theorem 
 In
view of equation (<ref>) with
γ=^ξ-k, we have

    Q(θ)+φ(t_^ξ-k(θ);θ)<1/k,    θ∈[1,θ^*_^ξ-k).

By continuity of the map
γ↦φ(t_γ(θ);θ), inequality
(<ref>) extends to

    Q(θ)+φ(t_^ξ-δ-k(θ);θ)<1/k,

for some δ>0 small enough.

According to the definition (<ref>), there exists an
integer N such that

    max_1≤ℓ≤ qinf_x∈ V_
    N^c_ℓξ̌_(1)(x)≥^ξ-δ.

For a specific reduction
_ℓξ̌(x)∈_ℓℝ̌^q, with components
_ℓξ̌_j(x)=ξ_j(x)-ξ_ℓ(x) (jℓ),
denote by _ℓF_i(u;θ) (iℓ) the
corresponding functions analogous to  F_i(u;θ)
that were defined in (<ref>) under the standard reduction
(i.e., with ℓ=q). Lemma <ref> (modified to the case of
reduction via the ℓ-th coordinate) implies that

    max_iℓsup_u∈
    B_ℓ(x)∇_ℓF_i(u;θ)_1≤
    Q(θ)+φ(t__ℓξ̌_(1)(x)-k(θ);θ),

where

    B_ℓ(x):=   {u∈_ℓℝ̌^q:
     min_jℓ u_j≥(_ℓξ̌_(1)(x)-k) lnθ}.

Furthermore, exploiting monotonicity and continuity of the function
t↦φ(t;θ), we obtain from (<ref>)

    min_1≤ℓ≤ qsup_x∈ V_ N^cmax_iℓsup_u∈
    B_ℓ(x)∇_ℓF_i(u;θ)_1
       ≤
    Q(θ)+φ(t^*_N(θ);θ),

with

    t^*_N(θ):=max_1≤ℓ≤ q inf_x∈ V_
    N^ct__ℓξ̌_(1)(x)-k(θ).

Due to the bound (<ref>), we have t^*_N(θ)≥
t_^ξ-δ-k(θ), and by monotonicity of
t↦φ(t;θ) it follows that

    Q(θ)+φ(t^*_N(θ);θ)≤
    Q(θ)+φ(t_^ξ-δ-k(θ);θ)<1/k,

according to the estimate (<ref>). Together with
(<ref>), this implies that (cf. condition (<ref>))

    λ_N(θ):=kmin_1≤ℓ≤
    qsup_x∈ V_ N^cmax_iℓsup_u∈
    B_ℓ(x)∇_ℓF_i(u;θ)_1 <1.

Hence, by an extended version of Theorem <ref> (see
Remark <ref>), it follows that the solution
g(x) to the functional equation (<ref>) is
unique on the subset {x∈ V_ N^c}. Finally, the values of
the solution g(x) for x∈ V_N are retrieved
uniquely by the “backward” recursion (<ref>)
using (<ref>). Thus, the proof of Theorem <ref> is
complete.




  §.§.§ Proof of Theorem 
 Let
μ_β,ξ and μ̅_β,ξ be two SGMs
determined by the functions g(x) and
g̅(x), respectively, each satisfying the
functional equation (<ref>). Our aim is to show that, under
the theorem's hypotheses,
g(x)≡g̅(x), which would imply
that μ_β,ξ=μ̅_β,ξ. The idea of
the proof is to obtain a suitable upper bound on the norm
g(x)-g̅(x)_∞ for x∈
W_n in terms of
g(y)-g̅(y)_∞ for y∈
W_n+1, 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
ℱ_0⊂ℱ_1⊂…ℱ_n⊂⋯ consisting of the sigma-algebras
ℱ_n generated by the values of the random field
ξ in the sequence of expanding balls V_n={x∈
V d(x_∘,x)≤ n},

    ℱ_n:=σ{ξ(x) x∈
    V_n},     n∈ℕ_0.

Put

    λ(θ):=k (Q(θ)
    +𝔼[φ(t_ξ̌_(1)(x)-k(θ);θ)]),

where the expectation does not depend on x∈ V due to the i.i.d.assumption on the field {ξ(x)}. Let us first show
that for each x∈ W_n (n≥ 1) we have the upper bound

    𝔼(g(x)-g̅(x)_∞|ℱ_n-1)≤ 2lnθ(λ(θ))^m,     m∈ℕ_0,

where 𝔼(·|ℱ_n-1) stands for the
conditional expectation.

Fix x∈ W_n. The base of induction (m=0) is obvious, noting
that, due to (<ref>), (<ref>) and Lemma <ref>,

    g(x)-g̅(x)_∞≤g(x)_∞+g̅(x)_∞≤
    2lnθ.

Suppose now that the bound (<ref>) is true for some
m∈ℕ_0, and show that it holds for m+1 as well. By
Lemma <ref> we have

    g(x)-g̅(x)_∞   =
    g(x)-g̅(x)_∞
       ≤max_1≤ i≤ q-1sup_u∈ B(x)∇
    F_i(u;θ)_1∑_y∈
    S(x)g(y)-g̅(y)_∞,

where B(x)⊂ℝ^q-1 is the ball of radius
klnθ centred at βξ̌(x),

    B(x):={u∈ℝ^q-1u-βξ̌(x) _∞≤
    klnθ}.

Recalling that β=lnθ, observe that if
u=(u_1,…,u_q-1)∈ B(x) then, for each
i=1,…,q-1,

    u_i≥βξ̌_i(x)-klnθ=lnθ(ξ̌_i(x)-k),

and hence

    min_1≤ i≤ q-1 u_i≥lnθmin_1≤ i≤ q-1(ξ̌_i(x)-k)=lnθ(ξ̌_(1)(x)-k),

with ξ̌_(1)(x)=min_1≤ i≤ q-1ξ̌_i(x).
Therefore, on applying Lemma <ref> we have

    max_1≤ i≤ q-1sup_u∈ B(x)∇
    F_i(u;θ)_1≤ Q(θ) +
    φ(t_ξ̌_(1)(x)-k(θ);θ).

Thus, returning to (<ref>) we get

    g(x)-g̅(x)_∞≤[Q(θ) +
    φ(t_ξ̌_(1)(x)-k(θ);θ)]∑_y∈
    S(x)g(y)-g̅(y)_∞.


Now, take the conditional expectation
𝔼(·|ℱ_n) on both sides of
(<ref>), noting that the factor in front of the sum is a
random variable measurable with respect to the sigma-algebra
ℱ_n, so it can be pulled out of the expectation (see
<cit.>). This yields

    𝔼[g(x)-g̅(x)_∞|ℱ_n]
       ≤[Q(θ) +
    φ(t_ξ̌_(1)(x)-k(θ);θ)]∑_y∈
    S(x)𝔼[g(y)-g̅(y)_∞|ℱ_n]
       ≤ k[Q(θ) +
    φ(t_ξ̌_(1)(x)-k(θ);θ)]
    · 2lnθ(λ(θ))^m,

where in the last inequality we used that S(x)=k and also
applied the induction hypothesis to each y∈ S(x)
(see (<ref>)). In turn, using the tower property of
conditional expectation (see <cit.>) with ℱ_n-1⊂ℱ_n,
from (<ref>) we obtain

    𝔼(g(x)-g̅(x)_∞|ℱ_n-1)
       =𝔼[𝔼[g(x)-g̅(x)_∞|ℱ_n]
    |ℱ_n-1]
       ≤ k 𝔼[Q(θ) +
    φ(t_ξ̌_(1)(x)-k(θ);θ)|ℱ_n-1] · 2lnθ(λ(θ))^m
       =k 𝔼[Q(θ) +
    φ(t_ξ̌_(1)(x)-k(θ);θ)]
    · 2lnθ(λ(θ))^m
       = 2lnθ(λ(θ))^m+1

(see (<ref>)). Thus, the induction step is completed
and, therefore, the claim (<ref>) is true for all m≥0. In
particular, again using the tower property of conditional
expectation, from (<ref>) we readily get

    𝔼(g(x)-g̅(x)_∞)
       =𝔼[𝔼(g(x)-g̅(x)_∞|ℱ_n-1)]
       ≤ 2lnθ(λ(θ))^m.


Now, if θ^†>1 is the (unique) solution of the
equation (<ref>), then λ(θ)<1 for
all θ∈[1,θ^†). Hence, taking the limit of
(<ref>) as m→∞ gives

    𝔼(g(x)-g̅(x)_∞)=0,    
    x x_∘,

and therefore g(x)=g̅(x) (a.s.)for any x x_∘. It remains to notice that this equality
extends to x=x_∘  by the recursion (<ref>).




§ ANALYSIS OF THE MODEL WITH CONSTANT FIELD



 §.§ Classification of positive solutions to the
system (<ref>)


Denote

    p_k(z):=z^k-1+…+z,

so that

    z^k-1=(z-1)(p_k(z)+1).


 Let (z_1,…,z_q-1) be a solution to
(<ref>), with z_i>0 i=1,…,q-1.


  (a)
If θ=1 then z_1=…=z_q-1=1 is the unique solution.


  (b) If θ>1 and α=0 then
either z_1=…=z_q-1=1 or there is a non-empty subset
ℐ_0⊆{1,…,q-1}, with
m:=ℐ_0 ranging from 1 to q-1, such that

    z_i= u   if  i∈ℐ_0,
    
    1   otherwise,

where u=u(θ,m)1 satisfies the equation

    1=(θ-1)(p_k(u)+1)/θ+m u^k+q-1-m.



  (c) If θ>1 and α 0 then



  (i)
either z_1=u and z_2=…=z_q-1=1, where u=u(θ,α)
satisfies the equation

    u=1+(θ-1)(θ^α u^k-1)/θ+θ^α
    u^k+q-2

and, in particular, u1


  (ii) or, provided that q≥3, there is a non-empty subset ℐ_1⊆{2,…,q-1}, with m:=ℐ_1 ranging from 1 to
q-2, such that

    z_i= u    if   i=1,
    
    v   if  i∈ℐ_1,
    
    1    otherwise,

where u= u(θ,α,m) and v= v(θ,α,m)1
satisfy the set of equations

    { u   =1+(θ-1)(θ^α
    u^k-1)/θ+θ^α u^k+m v^k+q-2-m,
    
    1   =(θ-1)(p_k(v)+1)/θ+θ^α u^k+m
    v^k+q-2-m,
    .

and, in particular, u1 and u v.




As a general remark, observe that z_i=1 solves the i-th
equation of the system (<ref>) regardless of all other z_j
with j i.


  (a) Obvious.


  (b) In this case, the system (<ref>) takes the form

    z_i=1+(θ-1)(z_i^k-1)/θ+∑_j=1^q-1z_j^k,    
    i=1,…,q-1.

Suppose that the set ℐ_0:={i≥1 z_i1} is
non-empty. By virtue of the identity (<ref>), for any
i∈ℐ_0 equation (<ref>) is reduced to

    (θ-1) p_k(z_i) =1+∑_j=1^q-1z_j^k.

Because the right-hand side of (<ref>) does not depend on
i∈ℐ_0 and the function p_k(z) is strictly
increasing for z>0, it follows that z_i=: u=
(i∈ℐ_0). Specifically, if ℐ_0=m≥1
then equation (<ref>) specializes to (<ref>).


  (c) The proof is similar to part (b). First of all, note that u:=z_11, for otherwise the first
equation in (<ref>)  is not satisfied unless θ = 1 or
α = 0, either of which is ruled out. Next, if
z_2=…=z_q-1=1 then the first equation for z_1=u in
(<ref>) specializes to (<ref>), as stated.

Suppose now that ℐ_1:={i≥ 2  z_i1}∅, then similarly as above we show that z_i=
(i∈ℐ_1), and the system (<ref>) specializes to
equations (<ref>) with z_1=u and z_i=v (i∈ℐ_1).

Finally, assuming to the contrary that u=v and comparing the
equations in (<ref>), we would conclude that θ^α
u^k=u^k, that is, α=0, which is ruled out. Hence, u v
as claimed.


Thus, the proof of Lemma <ref> is complete.



It is not hard to check that, in the limit as α→0, case (c)
of Lemma <ref> transforms into case (b).




 §.§ Proof of Theorem

 By the substitution

    u^k=(q-1) x/θ^α+1_,     
    x>0,

equation (<ref>) can be represented in
the form

    ax=f(x),     f(x):=(1+x/b+x)^k,

with the coefficients (cf. (<ref>))

    a= a(θ):=q-1/θ^k+1+α>0,     b=
    b(θ):=θ(θ+q-2)/q-1≥ 1.

Equation (<ref>) is well known in the theory of Markov chains on
the Cayley tree (see, e.g., <cit.> or
<cit.>), and it is easy to analyse the number of its
positive solutions. The case b=1 is obvious. Assuming b>1, it is
straightforward to check that f(x) is an increasing function, with
f(0)=b^-k<1 and lim_x→∞f(x)=1; also, it has one
inflection point x_0=1/2(b(k-1)-(k+1)), such
that f(x) is convex for x<x_0 and concave for x>x_0 (note that
x_0>0 only when b>k+1/k-1). Therefore, the equation
(<ref>) has at least one and at most three positive solutions.
In fact, by fixing b>0 and gradually increasing the slope a>0 of
the ray y=ax (x≥0), it is evident that there are more than one
solutions (i.e., intersections with the graph y=f(x)) if and only
if the equation x f'(x)=f(x) has at least one solution, each
such solution x=x_* corresponding to the line y=ax, with
a=f'(x_*), serving as a tangent to the graph y=f(x) at point
x=x_*. In turn, from (<ref>) we compute

    f'(x)=k(1+x/b+x)^k-1b-1/(b+x)^2=f(x) k(b-1)/(b+x)(1+x),

and it readily follows that the condition x f'(x)=f(x)
transcribes as the quadratic equation (<ref>),
with discriminant D given by (<ref>). Thus, if D>0, that
is, b>(k+1/k-1)^2, then the equation
(<ref>) has two distinct roots
0<x_-<x_+, corresponding to the “critical” values
a_±=f(x_±)/x_± (see (<ref>)
and (<ref>)). Furthermore, using (<ref>) it is easy
to see that the function x↦ f(x)/x is increasing on the
interval x∈[x_-,x_+]; hence, a_-<a_+.

To summarize, if b≤(k+1/k-1)^2 then the
equation (<ref>) has a unique solution, whereas if
b>(k+1/k-1)^2 then there are one, two or three
solutions according as a∉ [a_-,a_+], a∈{a_-,a_+} or
a∈ (a_-,a_+), respectively. Adapting these results to equation
(<ref>), in view of the second formula in (<ref>) the
condition b(θ)>(k+1/k-1)^2 is equivalent
to θ>θ_ c, with θ_ c=θ_
c(k,q) defined in (<ref>). The corresponding critical
values α_± of the field parameter α are determined
by the first formula in (<ref>), that is,

    θ^ k+1+α_±=q-1/a_∓,

leading to formula (<ref>). This completes the proof of
Theorem <ref>.



 §.§ Proof of Theorem 

For m∈{1,…,q-2}, denote by m':=q-1-m the
“conjugate” index, m'∈{1,…,q-2}. Recall the
notation (<ref>),

    L_m(v;θ):=(θ-1) p_k(v)-m v^k -m',    θ≥ 1,  v≥0,

where the polynomial p_k(v) is defined in (<ref>).



  (a)
For every θ>1, there is v_m= v_m(θ)>0 such that
the function v↦ L_m(v;θ) is increasing for 0<v<v_m
and decreasing for v> v_m, thus attaining its unique maximum value
at v=v_m,

    L_m^*(θ):=
    L_m(v_m(θ);θ)=max_v>0L_m(v;θ),    θ>1.



  (b) For each m≥ 1, the function θ↦ L_m^*(θ)
defined in (<ref>) is continuous and monotone increasing,
with lim_θ→∞L_m^*(θ)=∞. Furthermore,
L_m^*(θ) has a unique zero θ_m>1, that is,

    L_m^*(θ_m)=L_m(v_m(θ_m);θ_m)=0.


  (c) The value v_m^*:=v_m(θ_m) is the unique positive root of the
equation

    m∑_i=1^k-1i v^k-i-m'∑_i=1^k-1i v^i-k=0.

In particular, v_m^*=1 if m=1/2(q-1) and v_m^*>1 if
m<1/2(q-1).


(a) Differentiating (<ref>) with respect to v, we get

    ∂ L_m(v;θ)/∂ v   =(θ-1)
    p'_k(v)-km v^k-1
       =v^k-1((θ-1)∑_i=1^k-1k-i/v^i-km).

It is evident that the function in the parentheses in (<ref>)
is continuous and monotone decreasing in v>0, with the limiting
values +∞ as v↓0 and -km<0 as v→∞.
Hence, there is a unique root v_m=v_m(θ) of the equation
∂ L_m(v;θ)/∂ v=0, that is,

    (θ-1) p'_k(v_m)-km v_m^k-1 = 0,

and, moreover, ∂ L_m/∂ v>0 for 0<v<v_m and
∂ L_m/∂ v<0 for v>v_m. Thus, claim (a) is proved.

(b) Note that the derivative v'_m(θ) exists by the inverse
function theorem applied to equation (<ref>).
Differentiating (<ref>) and using (<ref>), we get

    L_m^*(θ)/θ   =∂
    L_m(v;θ)/∂ v|_v=v_m(θ)×
    v_m(θ)/θ+∂ L_m(v;θ)/∂θ|_v=v_m(θ)
       =p_k(v_m(θ))>0.

Thus, L_m^*(θ) is continuously differentiable and (strictly)
increasing.[The monotonicity of L_m^*(θ) is obvious
without proof, because the function θ↦ L_m(v;θ) is
monotone increasing for each v>0, since ∂
L_m/∂θ=p_k(v)>0.] Observe from (<ref>) that

    L_m^*(θ)
       ≥ L_m(v;θ)|_v=1
       =(θ-1) p_k(1)-m-m'
       =(θ-1)(k-1)-(q-1)→+∞,    θ→∞.

On the  other hand, from (<ref>) we see that if
1<θ<k/k-1 then, for all m≥1,

    ∂ L_m(v;θ)/∂
    v|_v=1   =(θ-1)∑_i=1^k-1(k-i)-km
       =(θ-1) (k-1) k/2-km
       <k(1/2-m)<0.

Therefore, by part (a), for such θ we have 0<v_m(θ)<1,
hence, for all m≤ q-2,

    L_m^*(θ)   =(θ-1) p_k(v_m)-m v_m^k-m'
       <(k/k-1-1) p_k(1)-m'
       =1-m'≤ 0.

Thus, combining (<ref>) and (<ref>), it follows that
there is a unique root θ=θ_m of the equation
L_m^*(θ)=0, which proves claim (b).

(c) Elimination of θ=θ_m from the system of equations
(<ref>) and (<ref>) gives for
v_m^*=v_m(θ_m) a closed equation,

    mk v^k-1p_k(v)-(m v^k+m') p'_k(v)=0,

which can be rearranged to a more symmetric form (<ref>). The
uniqueness of the root v^*_m is obvious, because the left-hand
side of (<ref>) is a continuous, increasing function in v>0,
with the range from -∞ to +∞. Finally, observe that
for v=1 the left-hand side of (<ref>) is reduced to
(m-m')· k(k-1)/2, which vanishes if m=m' and is negative
if m<m', so that, respectively, v_m^*=1 or v_m^*>1, as
claimed.

Thus, the proof of Lemma <ref> is complete.



The statements of Lemma <ref> including the identity
(<ref>) are valid with a continuous parameter m.


We can now proceed to the proof of Theorem <ref>. Assume that
α0. The second equation in the system (<ref>) is
reduced to

    θ+θ^α u^k+m v^k +m'-1=(θ-1)(p_k(v)+1),

which can be rewritten, using  the notation (<ref>), in the
form

    θ^α u^k=L_m(v;θ).

Furthermore, substituting (<ref>) and (<ref>) into the
denominator and numerator, respectively, of the ratio in the first
equation of (<ref>), we get

    u=p_k(v)+L_m(v;θ)/p_k(v)+1.

Finally, substituting (<ref>) back into (<ref>), we
obtain the equation

    θ^α=K_m(v;θ),

where (cf. (<ref>))

    K_m(v;θ):=L_m(v;θ)(p_k(v)+1)^k/(p_k(v)+L_m(v;θ))^k.

Conversely, all steps above are reversible, so equations
(<ref>) and (<ref>) imply the system (<ref>).

Note from (<ref>) that v>0 must satisfy the condition
K_m(v;θ)>0, that is, v∈𝒱_m^+(θ)
(see (<ref>)); by Lemma <ref>(b), this is possible
if and only if θ>θ_m. Moreover, the
equation (<ref>) has a solution v>0 if and only if α≤α_m(θ), with the critical threshold α_m(θ)
defined in (<ref>). This completes the proof of
Theorem <ref>.



 §.§ Proof of Theorem <ref>

Recall that the critical point θ̃_1 was defined
in (<ref>). If q=2 then the only solutions of the
compatibility system (<ref>) are provided by equation
(<ref>); therefore, Theorem <ref>(a) readily
follows from Theorem <ref>.

More generally (i.e., for q≥ 3), in order that
ν(θ,α)≥2, either there must be at least two
solutions of equation (<ref>), that is, (θ,α)∈
A_q (see Theorem <ref>), or, since we always have
ν_0(θ,α)≥1, there should exist at least one solution
(u,v) of the system (<ref>). By Theorem <ref>, such
solutions exist if α≤α_m(θ) for some m; since
α_1(θ) is a majorant of the family
{α_m(θ)} (see Proposition <ref>),
the latter condition is reduced to α≤α_1(θ),
which leads to the inclusion (θ,α)∈ B_q. However, we
must ensure that this solution also satisfies the constraint v1
(see (<ref>)). By Lemma <ref>, this is certainly
true if m=1<1/2(q-1), that is, q>3, which proves
Theorem <ref>(c).

Finally, Theorem <ref>(b) (for q=3) readily follows by
the next lemma about the maximum of the function v↦
K_1(v;θ) over the domain v∈𝒱_1^+(θ)
(see (<ref>)).


Let q=3 and k≥2.  
  (a) For all
θ>θ̃_1, we have K_1(v;θ)|_v=1<max_v∈𝒱_1^+(θ)
K_1(v;θ).


  (b)
Let k∈{2,3,4}. If 1<θ≤θ̃_1 then the
function v↦ K_1(v;θ) has the unique maximum at v=1,
that is, K_1(v;θ)<K_1(1;θ) for any v1.


The proof of the lemma is elementary but tedious, so it is deferred
to Appendix <ref>.


The maximum of the function v↦ K_m(v;θ), as well as the
number of solutions v>0 of the equation (<ref>) for various
values of parameters are illustrated in Figure <ref> (for the
regular case q≥4) and in Figure <ref> (for the special case
q=3).







As mentioned in Remark <ref>, the case q=3 is
truly critical with regard to the uniqueness. Recall that
ν(θ,α) denotes the number of positive solutions of the
system (<ref>); the function α_1(θ) is defined
in (<ref>) and θ_1^0 is its zero. The proof of the
next proposition relies on some lemmas that will be proved later, in
Sections <ref> and <ref>.

Let q=3 and k≥2. There exists ε>0 small
enough such that


  (a) ν(θ,α_1(θ))=1 if 1≤θ<θ_1+ε,


  (b) ν(θ,α_1(θ))≥ 2 for
all θ>θ^0_1-ε.


(a) As shown in the proof of Lemma <ref> (see
Appendix <ref>), ∂^2 K_1/∂ v^2|_v=1<0
for all θ∈[θ_1,θ̃_1); in particular,
∂^2 K_1(θ_1,v)/∂ v^2|_v=1<0. By
Lemma <ref>(c), the set 𝒱_1^+(θ)
(see (<ref>)) is reduced for θ=θ_1 to the single
point v=1, and
K_1^*(θ_1)=max_v∈𝒱^+_1(θ_1)K_1(v;θ_1)=K_1(1;θ_1)=0.
By continuity, it follows that  ∂^2 K_1(θ;v)/∂
v^2<0 for each θ∈[θ_1,θ_1+ε) and
all v∈𝒱^+_1(θ); that is, the function v↦
K_1(v;θ) is concave on 𝒱^+_1(θ) and therefore
has a unique maximum located at v=1 (remembering that ∂
K_1/∂ v|_v=1=0). But the solution (u,v) with v=1
is not admissible (see (<ref>)), hence
ν(θ,α_1(θ))=1, with the only solution of
(<ref>) coming from equation (<ref>).

(b) We have α_1(θ_1^0)=0, hence K_1^*(θ_1^0)=1
(see (<ref>)) and, by Lemma <ref>(b), also
L_1^*(θ^0_1)=1. If v_1=v_1(θ_1^0) is the point where
the latter maximum is attained, that is,
L_1(v_1;θ_1^0)=L_1^*(θ_1^0)=1, then it holds that
v_1>1 (see Lemma <ref> below). Furthermore, according to
Lemma <ref>(a), v_1 is the unique maximum of the
function v↦ L_1(v;θ_1^0), and in particular
L_1(1;θ_1^0)<L_1^*(θ_1^0)=1.

From the definition (<ref>), it follows that also
K_1(v_1;θ_1^0)=1=K_1^*(θ_1^0),[By the scaling
property (<ref>), we also have
K_1(v_1^-1;θ_1^0)=K_1(v_1;θ_1^0)=1.] and since
L_1(1;θ_1^0)<1, Lemma <ref>(a) implies that
K_1(1;θ_1^0)<1=K_1^*(θ_1^0). Thus, the corresponding
solution of the system (<ref>) satisfies the
condition (<ref>), and therefore
ν(θ_1^0,α_1(θ_1^0))≥ 2.

By continuity of K_1(1;θ) and K_1^*(θ), for all
θ∈(θ_1^0-ε,θ_1^0] (with
ε>0 small enough) we still have that
K_1(1;θ)<K_1^*(θ), so the maximum is attained outside
v=1. Thus, by the same argument as before, the claim follows.





 §.§ Proof of Theorem <ref>

(a) First, let q=2. According to Lemma <ref>, the system
(<ref>) is reduced to the single equation (<ref>), and
by Theorem <ref> the number of its solutions is not more than
3=2^2-1; furthermore, for θ>θ_
c=k+1/k-1 and
α_-(θ)<α<α_+(θ), there are exactly
three solutions, so the upper bound is attained.

(b) Let now α=0 (and q≥ 3). Due to Lemma <ref>(b),
either z_i≡ 1 or the system (<ref>) is reduced to
the equation (<ref>) indexed by m=1,…,q-1, which can be
rewritten as L_m(u;θ)=1 (see (<ref>)). By
Lemma <ref>(a), the latter equation has no more than two
roots. Hence, considering permutations of the values u1 over
the q-1 places, it is clear that the total number of solutions to
(<ref>) is bounded by

    1+2 ∑_m=1^q-1q-1m=1+2(2^q-1-1)=2^q-1.

Moreover, for θ>1 large enough, there will be exactly two
roots of each of the equations L_m(u;θ)=1, because
L_m^*(θ)=max_u>0L_m(u;θ)→∞ as
θ→∞ (see Lemma <ref>(b)). Therefore, the
upper bound (<ref>) is attained.

(c) Finally, let k=2 and α0. First of all, up to three
solutions of the system (<ref>) arising from the
equation (<ref>) are ensured by Lemma <ref> (see also
Theorem <ref>). Other solutions are determined by the system
(<ref>) indexed by m=1,…,q-2, which in turn depends on the
solvability of the equation (<ref>). In the case k=2, the
latter is a polynomial equation of degree 4, and therefore has at
most four roots v>0, for each m. The value u>0 is then
determined uniquely by formula (<ref>), and it occupies the
first place in the vector z=(z_1,…,z_q-1). As
for the root v>0, it occupies m out of the q-2 remaining
places. Counting the total number of such permutations, we get the
upper bound

    3+4 ∑_m=1^q-2q-1m=3+4(2^q-2-1)=2^q-1,

as required. This completes the proof of Theorem <ref>.



§ FURTHER PROPERTIES OF THE CRITICAL CURVES
Α_±(Θ) AND Α_M(Θ)




 §.§ Properties of α_±(θ)



The quantities a_±, defined in (<ref>) for
b≥(k+1/k-1)^2, satisfy the identity

    a_+ × a_-=b^-k-1.


Using (<ref>) and noting that x_-=b/x_+ (see
equation (<ref>)), we find

    a_-   =1/x_-(1+x_-/b+x_-)^k
       =x_+/b(1+b/x_+/b+b/x_+)^k
       =x_+/b^k+1(b+x_+/1+x_+)^k
       =b^-k-1/a_+,

and formula (<ref>) follows.



Suppose that b≥(k+1/k-1)^2. Then the
following inequalities hold,

    a_+<b^-1,     a_->b^-k.


From (<ref>), for all x≥0 we get the upper bound

    f'(x)
       =(1+b-1/1+x)^-kk(b-1)/(b+x)(1+x)<
    1/b+x≤1/b,

noting that, by Bernoulli's inequality,

    (1+b-1/1+x)^k
    >k(b-1)/1+x.

Hence, a_+=f'(x_+)<b^-1, and the first inequality in
(<ref>) is proved. The second inequality then readily follows
from the identity (<ref>).



The functions α_±(θ) satisfy the following identity,

    α_-(θ)+α_+(θ)=2ln(q-1)+(k+1)(ln
    b(θ)-2)/lnθ,    θ≥θ_ c,

where b(θ) is defined in (<ref>). In particular, if
q=2 then α_-(θ)+α_+(θ)≡ 0 for all
θ≥θ_ c=k+1/k-1.

Using (<ref>), we obtain

    θ^2(k+1)+α_-(θ)+α_+(θ)=(q-1)^2/a_-(θ)
    a_+(θ),    θ≥θ_ c,

and the identity (<ref>) follows upon substituting
formula (<ref>).


 The
functions α_±(θ) [θ_
c,∞)→ℝ defined in (<ref>) have the
following “boundary” values,

    α_±(θ_ c)   =-(k+1)+1/lnθ_ c(ln(q-1)+(k+1)lnk+1/k-1),
    α_±(θ)   →± (k-1)     (θ→∞).

In particular, α_±(θ_ c)=0 if q=2 and
α_±(θ_ c)>0 if q>2.

 Recall from the proof of
Theorem <ref> (see Section <ref>) that the critical
value θ=θ_ c corresponds to b(θ_
c)=(k+1/k-1)^2, whereby the quadratic equation
(<ref>) has the double root

    x_±(θ_ c)=√(b(θ_ c))=k+1/k-1.

Hence, using (<ref>), we find

    a_±(θ_ c)=(k-1/k+1)^k+1,

which, together with (<ref>), yields
formula (<ref>).

If q=2 then formula (<ref>) gives θ_
c=k+1/k-1, and it readily follows from
(<ref>) that α(θ_ c)=0. For q>2,
using the relation (<ref>) observe that the required
inequality α_±(θ_ c)>0 is reduced to

    θ_ c^k+1<q-1/a_±(θ_
    c)=(k+1/k-1)^k+1(q-1),

that is,

    θ_ c<(k+1/k-1)(q-1)^1/(k+1).

Denote

    ρ_k:=k+1/k-1>1,     s:=(q-1)^1/(k+1)>1,

then (<ref>) takes the form

    θ_ c<ρ_k s.

Furthermore, recalling that b(θ_
c)=(k+1/k-1)^2=ρ_k^2 and b(θ) is
monotone increasing for θ>1 (see (<ref>)
and (<ref>)), the inequality (<ref>) is equivalent
to

    ρ_k^2=b(θ_ c)<b(ρ_k s)=ρ_k
    s(ρ_k s+s^k+1-1)/s^k+1,

that is,

    s^k+1-1>ρ_k (s^k-s),

which is reduced, upon dividing by s-1>0 and substituting
ρ_k-1=2/k-1, to

    ϕ_k(s):=s^k+1-2 p_k(s)/k-1>0.

In fact, it is easy to show that ϕ_k(s)>0 for any s>1.
Indeed, since p_k(1)=k-1, we have ϕ_k(1)=0, while

    ϕ_k'(s)   =k s^k-1-2/k-1∑_i=1^k-1i
    s^i-1
       =s^k-1(k-2/k-1∑_i=1^k-1i s^-(k-i))
       >s^k-1(k-2/k-1∑_i=1^k-1i)
       =s^k-1(k-2/k-1·k(k-1/2)=0.

Thus, inequality (<ref>) is verified, which implies that
α(θ_ c)>0, as argued above.

Let us now prove (<ref>). Using the definition of b=b(θ)
and D=D(θ) (see (<ref>) and (<ref>), respectively),
we obtain the following asymptotics as θ→∞,

    b=θ^2/q-1+O(θ),     √(D)=θ^2(k-1)/q-1+O(θ),

and

    x_+=θ^2(k-1)/q-1+O(θ),     x_-=b/x_+
    =1/k-1 +O(θ^-1).

Hence,

    ln x_±=(1±1)lnθ +O(1),    lnb+x_±/1+x_±=(1∓1)lnθ+O(1).

Using formula (<ref>), this yields

    -ln a_±(θ)   =ln
    x_±+klnb+x_±/1+x_±
       =((k+1)∓ (k-1))lnθ+O(1).

Therefore, from (<ref>) we get

    k+1+α_±=-ln a_∓(θ)/lnθ+o(1)=(k+1)±
    (k-1)+o(1),

and the limit (<ref>) follows.



The functions α_±(θ) satisfy the following bounds,

α_-(θ)   >-(k-1),       θ   ≥θ_c,

 α_+(θ)   <k-1,       θ   ≥max{θ_c, θ̅},

where

    θ̅=θ̅(k,q):=q-2/(q-1)^(k-1)/k-1,     q>2,
     1,   q=2.


Using the relation (<ref>), the first
inequality in (<ref>) and the definition of b
in (<ref>), we get

    θ^ k+1+α_-=
    q-1/a_+   >(q-1) b
       =θ(θ+q-2)≥θ^2,     q≥2,

which proves the bound (<ref>).

Similarly, using the second inequality in (<ref>) we have

    θ^ k+1+α_+=q-1/a_-   <(q-1)
    b^k
       =(q-1)^1-kθ^2k(1+q-2/θ)^k.

Noting from (<ref>) that, for θ≥θ̅,

    1+q-2/θ≤
    1+q-2/θ̅=(q-1)^(k-1)/k,     q≥ 2,

it is easy to see that the right-hand side of
(<ref>) is bounded above by θ^2k; hence,
the inequality (<ref>) follows.



Note that θ̅>1 for q≥ 3. Of course, if θ_
c≥θ̅ then the upper bound (<ref>) holds
for all θ≥θ_ c; however, the ordering between
θ_ c and θ̅ depends on k and q. For
example, for k=2 and q=5

    θ̅=3<θ_ c≐ 4.6847,

whereas for k=2 and q=50

    θ̅=8>θ_ c≐ 7.8904.

In fact, for large q the upper bound (<ref>) fails near
θ_ c; indeed, using (<ref>) we get

    θ_
    c=q-2/2(√(1+4(q-1)/(q-2)^2(k+1/k-1)^2)-1)∼(k+1/k-1)^2,     q→∞,

and, according to (<ref>),

    α_±(θ_ c)∼ln q/lnθ_
    c→∞,     q→∞.




The function α_-(θ) is monotone decreasing for all
θ≥θ_ c, whereas α_+(θ) is
decreasing for θ≤θ^+_0 and increasing for
θ≥θ^+_0, with the unique minimum
α_+(θ^+_0)=0 at the critical point

    θ^+_0=θ^+_0(k,q):=1+q/k-1.

In the case q=2, we have θ_
c=θ_0^+=k+1/k-1 and, by Lemma <ref>,
α_+(θ)≡ - α_-(θ); hence, the function
α_+(θ) should be monotone increasing for all
θ≥θ_ c.

This conjecture is supported by computer plots (see
Figure <ref>). Towards a proof, we have been able to
characterize the unique zero θ_0^- of α_-(θ)
and to show rigorously that α_+(θ^+_0)=0 and
α'_+(θ^+_0)=0 (see
Proposition <ref>(b)), but the monotonicity
properties are more cumbersome to verify.


Note that the value (<ref>) coincides with a
known critical point in the case α=0, above which the
solution z=1 is unstable (see
<cit.>). Our
Proposition <ref>(b-i) explains the emergence of this
critical point and its explicit value (<ref>).






 §.§ Properties of L_m(v;θ) and K_m(v;θ)


Here and below, we assume that q≥3. Recall that
v_m=v_m(θ) is the unique maximum of the function v↦
L_m(v;θ) (see Lemma <ref>(a)), and θ_m is
defined by the relation (<ref>), where 1≤ m≤ q-2.


All results in this section hold true for a continuous
parameter m (cf. Remark <ref>), which is evident by
inspection of the proofs.


The next lemma describes the useful scaling properties of the
functions L_m(v;θ) and K_m(v;θ) (see (<ref>)
and (<ref>)) under the conjugation m↦ m'.

For each m=1,…,q-2 and m'=q-1-m, the following identities
hold for all v>0 and θ>θ_m,

    L_m'(v;θ)   =v^kL_m(v^-1;θ),
     K_m'(v;θ)   =K_m(v^-1;θ).

 Recalling the notation (<ref>), note
that

    p_k(v^-1)=v^-(k-1)+…+v^-1=v^-kp_k(v),

and similarly

    p_k(v^-1)+1=v^-(k-1)+…+v^-1+1 
    =v^-k+1(p_k(v)+1).

Using (<ref>) and (<ref>), we can write

    L_m(v^-1;θ)   =(θ-1) p_k(v^-1)-m v^-k -m'
       =v^-k((θ-1) p_k(v)-m-m' v^k)
       =v^-kL_m'(v;θ),

and formula (<ref>) follows. Furthermore, using
(<ref>) and substituting (<ref>) and
(<ref>), we obtain

    K_m(v^-1;θ)   =L_m(v^-1;θ)(p_k(v^-1)+1)^k/(p_k(v^-1)+L_m(v^-1;θ))^k
       =v^-kL_m'(v;θ)· v^-(k-1)
    k(p_k(v)+1)^k/(v^-kp_k(v)+v^-kL_m'(v;θ))^k
       =L_m'(v;θ) (p_k(v)+1)^k/(p_k(v)+L_m'(v;θ))^k
       =K_m'(v;θ),

which proves formula (<ref>).



For θ>θ_m, let v_m=v_m(θ) be the point where the
function v↦ L_m(v;θ) attains its (positive) maximum
value, that is,
L_m(v_m;θ)=max_v∈𝒱_m^+(θ)L_m(v;θ)
(see Lemma <ref>). The next result provides a strict
lower bound for v_m (cf. Lemma <ref>(c) for
θ=θ_m).


For all m in the range 1≤ m≤1/2(q-1), we have

    v_m(θ)>1,    θ>θ_m.



To the contrary, suppose first that v_m<1 for some
θ>θ_m. Then, according to Lemma <ref>, we
have

    L_m(v_m^-1;θ)=v_m^-kL_m'(v_m;θ)>L_m'(v_m;θ),

and furthermore,

    L_m'(v_m;θ)   =(θ-1) p_k(v_m)-m'v_m^k - m
       = L_m(v_m;θ)+ (m'-m)(1-v_m^k)≥ L_m(v_m;θ),

since m'=q-1-m≥ m by the hypothesis of the lemma. Combining
(<ref>) and (<ref>), we see that
L_m(v_m^-1;θ)>L_m(v_m;θ), which contradicts the
assumption that L_m(v_m;θ) is the maximum value of the
function v↦ L_m(v;θ).

Assume now that v_m=1 for some θ>θ_m. Then

    .∂ L(v;θ)/∂
    v|_v=v_m=1=(θ-1) p'_k(1)-k
    m=(θ-1)k(k-1)/2-k m=0,

whence (θ-1)(k-1)=2 m. Hence,

    L_m(1;θ)=(θ-1)(k-1)-(q-1)=2 m-(q-1)≤ 0,

which contradicts the assumption L_m(v_m;θ)>0.

Thus, the inequality (<ref>) is proved.


For θ>θ_m, let w_m=w_m(θ) be the point where the
function v↦ K_m(v;θ) attains its
positive maximum value, that is,
K_m(w_m;θ)=max_v∈𝒱_m^+(θ)K_m(v;θ).
Note that w_m(θ_m)=v_m(θ_m)=v_m^* (see
Lemma <ref>(c)). The importance of the next technical
lemma is pinpointed by involvement of the expression
p_k(v)-(k-1) L_m(v;θ) in the partial derivative
∂ K_m/∂ v (see formula (<ref>) below).
 Let 1≤ m≤1/2(q-1) and θ>θ_m.


  (a)
Let w∈𝒱_m^+(θ) be a critical point of
the function v↦ K_m(v;θ), that is, any solution of the
equation ∂ K_m/∂ v=0. Assume that either
 w<v_m and L_m(w;θ)<1, or 
w≥ v_m. Then

    p_k(w)-(k-1) L_m(w;θ)>0.


  (b)
In particular, the inequality (<ref>) holds for
w=w_m, that is,

    p_k(w_m)-(k-1) L_m(w_m;θ)>0.




(a) From the definition (<ref>), compute the partial
derivative

    ∂ K_m/∂
    v=(p_k+1)^k-1/(p_k+L_m)^k+1((p_k+1)(p_k-(k-1) L_m)∂
    L_m/∂ v-k L_m(1-L_m) p'_k ),

with the shorthand notation p_k:=p_k(v) and L_m:=L_m(v;θ).
Hence, the condition ∂ K_m/∂ v=0 is reduced to the
equality

    (p_k+1)(p_k-(k-1) L_m)∂ L_m/∂
    v-k L_m(1-L_m) p'_k = 0.

If w<v_m then ∂ L_m/∂ v|_v=w>0, and the
required inequality (<ref>) readily follows from
equation (<ref>) using that L_m(w;θ)<1. Similarly, if
w>v_m then ∂ L_m/∂ v|_v=w<0 and
equation (<ref>) implies the inequality (<ref>)
provided that L_m(w;θ)>1. Alternatively, if L_m(w;θ)≤
1 then, noting that w>v_m>1 (by Lemma <ref>), we obtain,
in agreement with (<ref>),

    p_k(w)>p_k(1)=k-1≥ (k-1) L_m(w;θ),

because the function w↦ p_k(w) is strictly increasing and
p_k(1)=k-1.

Lastly, if w=v_m then ∂ L_m/∂ v|_v=v_m=0 and
equation (<ref>) implies L_m(v_m;θ)=1. Again using
Lemma <ref>, similarly to (<ref>) we get

    p_k(w)=p_k(v_m)>p_k(1)=k-1=(k-1) L_m(w;θ),

so the inequality (<ref>) holds in this case as well.

(b) Let w=w_m be the point of maximum of the function v↦
K_m(v;θ). According to part (a), we only have to consider the
case where w_m<v_m and L_m(w_m;θ)≥ 1.

If L_m(w_m;θ)>1, let w>v_m>w_m be such that
L_m(w;θ)=L_m(w_m;θ), then

    K_m(w;θ)   =L_m(w;θ)(1-L_m(w;θ)-1/p_k(w)+L_m(w;θ))^k
       =L_m(w_m;θ)(1-L_m(w_m;θ)-1/p_k(w)+L_m(w_m;θ))^k
       >L_m(w_m;θ)(1-L_m(w_m;θ)-1/p_k(w_m)+L_m(w_m;θ))^k
       =K_m(w_m;θ).

Thus, K_m(w;θ)>K_m(w_m;θ), which contradicts the
assumption that v=w_m provides the maximum value of the function
v↦ K_m(v;θ).

Lastly, suppose that, for some θ>θ_m,

    L_m(w_m;θ)=1.

In view of the definition (<ref>), condition (<ref>)
implies that K_m(w_m;θ)=1. Let us prove that in this case we
must have w_m>1, which would then automatically imply the required
inequality (<ref>) (cf. (<ref>)
and (<ref>)). To the contrary, assume that w_m≤1. If
w_m=1 then, using the definition (<ref>) and recalling that
m+m'=q-1, the equation (<ref>) is reduced to

    L_m(1;θ)=(θ-1)(k-1)-(q-1)=1,

whence we find

    θ=1+q/k-1.

Furthermore, noting that

    p_k'(1)=∑_i=1^k-1 i=k(k-1)/2

and substituting (<ref>), from (<ref>) we get

    ∂ L_m(1;θ)/∂
    v=(θ-1)k(k-1)/2-k
    m=k(q/2-m)>0,

since m≤1/2(q-1)<q/2. Thus, w_m=1 is the left root
of the equation L_m(v;θ)=1. Denote by v̅>1 the
right root, that is, L_m(v̅;θ)=1 and ∂
L_m/∂ v|_v=v̅<0. It follows that
K_m(v̅;θ)=1 (see (<ref>)), so the maximum value 1
of the function v↦ K_m(v;θ) is also attained at
v=v̅>1. Returning to formula (<ref>), observe that

    ∂ K_m(v̅;θ)/∂ v
    =p_k(v̅)-(k-1)/p_k(v̅)+1×∂
    L_m(v̅;θ)/∂ v<0,

because p_k(v̅)>p_k(1)=k-1 and, as mentioned above, ∂
L_m/∂ v|_v=v̅<0. But the inequality
(<ref>) implies that there are points v<v̅ such that
K_m(v;θ)>K_m(v̅;θ), a contradiction. Hence, the
case w_m=1 is impossible.

Now, suppose that w_m<1. Then p_k(w_m)<p_k(1)=k-1 and, in view
of the condition (<ref>), from equation (<ref>) it
readily follows that ∂ L_m/∂ v|_v=w_m=0, that
is, L_m(w_m;θ)=1 is the maximum value of the function
v↦ L_m(v;θ). Hence, for all v<w_m we have

    L_m(v;θ)<1,    ∂ L_m(v;θ)/∂ v>0.

On the other hand, by (<ref>) and monotonicity of p_k(v),

    p_k(w_m)-(k-1)
    L_m(w_m;θ)   =p_k(w_m)-(k-1)
       <p_k(1)-(k-1)=0.

By continuity of the functions v↦ p_k(v) and v↦
L_m(v;θ), the inequality (<ref>) is preserved for all
v<w_m close enough to w_m:

    p_k(v)-(k-1) L_m(v;θ)<0.

Using (<ref>) and (<ref>), from (<ref>) it
follows that for such v we have ∂ K_m/∂ v<0. But
this means that the function v↦ K_m(v;θ) is
decreasing in the left vicinity of w_m, and thus w_m
cannot be a maximum, in contradiction with our assumption. Thus, we
have proved that w_m>1 as required, which completes the proof of
Lemma <ref>.


The next two lemmas provide useful bounds on w_m=w_m(θ).
First, there is a simple uniform upper bound.

For all m∈[1,q-2],

    w_m<θ,    θ≥θ_m.


Observe, using the definition (<ref>), that

    L_m(v;θ)|_v=θ   =(θ-1) p_k(θ)-mθ^k -m'
       =(θ-1)(θ^k-1+…+θ)-mθ^k -m'
       =-(m-1)θ^k-θ-m'<0,

and also (cf. (<ref>))

    .∂ L_m(v;θ)/∂ v|_v=θ   =(θ-1) p'_k(θ)-kmθ^k-1
       =(θ-1)((k-1)θ^k-2+…+2θ+1)-kmθ^k-1
       =-(km-k+1)θ^k-1-θ^k-2-…-θ-1<0.

Hence, the point v=θ lies to the right of the set
𝒱_m^+(θ)={v>0 L_m(v;θ)>0}
(see (<ref>)). But w_m∈𝒱_m^+(θ) and
therefore w_m<θ, as claimed in (<ref>).


The important lower bound for w_m=w_m(θ) is established next.



  (a)
For all m in the range 1≤ m<1/2(q-1), we have

    w_m>1,    θ>θ_m.


  (b)
If m=1/2(q-1) then the maximum point w_m which
may not be unique can be chosen so that w_m≥1.


If L_m(w_m;θ)≥1 then, by the inequality
(<ref>) of Lemma <ref>,

    p_k(w_m)>(k-1) L_m(w_m;θ)≥ k-1=p_k(1),

which implies, due to the monotonicity of p_k(·), that
w_m>1, in line with (<ref>). Thus, it remains to consider
the case L_m(w_m;θ)<1.

Assume first that w_m=1 for some θ>θ_m. Using the
definition (<ref>) and the value p_k(1)=k-1, we have

    L_m(1;θ)   =(θ-1) p_k(1)-m-m'
       =(θ-1)(k-1)-(q-1),

and also, recalling formula (<ref>),

    ∂ L_m(1;θ)/∂ v   =(θ-1)
    p'_k(1)-k m
       =k/2((θ-1)(k-1)-2 m)
       =k/2(L_m(1;θ)+m'-m).

Substituting (<ref>), (<ref>) and (<ref>) into
(<ref>), it is easy to check that the condition ∂
K_m/∂ v|_v=1=0 (see (<ref>)) is reduced to

    (1-L_m(1;θ))(m'-m)=0.

Since L_m(1;θ)=L_m(w_m;θ)<1 by assumption, the condition
(<ref>) is only satisfied if m'-m=0, that is,
m=1/2(q-1). Conversely, if m=1/2(q-1) (i.e., m'=m)
then, by the scaling formula (<ref>) of
Lemma <ref>, we have the identity

    K_m(v^-1;θ)=K_m'(v;θ)=K_m(v;θ),    θ>θ_m,v>0,

which implies that the maximum point w_m=w_m(θ) can always be
chosen so as to satisfy the inequality w_m≥1, which proves
part (b) of the lemma.

Finally, let m<1/2(q-1) and w_m<1 for some
θ>θ_m. Denote L_m^*(θ):= L_m(v_m(θ);θ)=
max_v∈𝒱_m^+(θ) L_m(v;θ) (see
Lemma <ref> and the definition (<ref>)). We need to
distinguish between two subcases, (i) L_m^*(θ)≤ 1 and (ii)
L_m^*(θ)>1, which require a different argumentation.

(i) Assuming first that L_m^*(θ)≤1, we will show that then

    K_m(w_m^-1;θ)>K_m(w_m;θ),

which would contradict the assumption that K_m(w_m;θ) is the
maximum value. By formula (<ref>) of
Lemma <ref>, we have
K_m(w_m^-1;θ)=K_m'(w_m;θ). Hence, recalling the
definition (<ref>) of the function K_m(v;θ), the
inequality (<ref>) is reduced to

    L_m'(w_m;θ)/(p_k(w_m)+L_m'(w_m;θ))^k>L_m(w_m;θ)/(p_k(w_m)+L_m(w_m;θ))^k.

Note that (cf. (<ref>))

    L_m'(w_m;θ)   =(θ-1) p_k(w_m)-m'w_m^k - m
       = L_m(w_m;θ)+ (m'-m)(1-w_m^k)
       > L_m(w_m;θ).

Hence, for the proof of the inequality (<ref>), it suffices
to show that the function L↦ L(p_k(w_m)+L)^-k is
strictly increasing on the interval
L∈[L_m(w_m;θ),L_m'(w_m;θ)]. Computing the
derivative of this function, we see that the claim holds provided
that

    p_k(w_m)/k-1> L,     L_m(w_m;θ)≤ L≤
    L_m'(w_m;θ),

or simply if

    p_k(w_m)/k-1>L_m'(w_m;θ).


The inequality (<ref>) is easy to prove. Indeed, using the
assumption w_m<1, observe from the definition (<ref>) that

    p_k(w_m)/k-1> w_m^k-1>w_m^k.

On the other hand, according to the scaling formula
(<ref>) of Lemma <ref>, we have

    L_m'(w_m;θ)=w_m^k L_m(w_m^-1;θ)≤ w_m^k
    L_m^*(θ)≤ w_m^k,

by virtue of the assumption L^*_m(θ)≤1. Now, the required
inequality (<ref>) readily follows from the estimates
(<ref>) and (<ref>).

Thus, the inequality (<ref>) is proved, and therefore the
assumptions w_m<1 and L_m^*(θ)≤ 1 are incompatible.

(ii) Assume now that L_m^*(θ)=L_m(v_m;θ)>1 and, as
before, w_m=w_m(θ)<1. Denote

    W_m= W_m(θ):={w>1.∂
    K_m(v;θ)/∂ v|_v=w=0},

that is, the set of all critical points of the function v↦
K_m(v;θ) (i.e., satisfying the equation (<ref>)) that
lie to the right of point v=1. By assumption,

    K_m(w_m;θ)>max_w∈ W_m K_m(w;θ),

and our aim is to show that this leads to a contradiction.

Since ∂ L_m/∂ v|_v=v_m=0 and
L_m(v_m;θ)>1, formula (<ref>) implies ∂
K_m/∂ v|_v=v_m>0 and, therefore, there is at least one
critical point w>v_m, which then automatically belongs to the set
W_m, because v_m>1 by Lemma <ref>. There may also be
critical points w∈ W_m such that 1<w<v_m; for these we may
assume, without loss of generality, that L_m(w;θ)<1, for
otherwise we consider the point w>v_m>w such that
L_m(w;θ)=L_m(w;θ), and it follows (similarly to
the derivation of inequality (<ref>)) that
K_m(w;θ)≥ K_m(w;θ), which means that such w
can be removed from the set W_m without affecting the maximum
in (<ref>).

Now, the idea is to increase the index m. Namely, treating m as
a continuous parameter (see Remark <ref>), differentiate
the function m↦ K_m(w_m(θ);θ) to obtain

    ∂ K_m(w_m;θ)/∂ m   =∂
    K_m(v;θ)/∂ v|_v=w_m×∂
    w_m(θ)/∂ m
         +∂ K_m(w_m;θ)/∂
    L|_L=L_m(w_m;θ)×∂
    L_m(v;θ)/∂ m|_v=w_m
       =(p_k(w_m)+1)^k(p_k(w_m)-(k-1)
    L_m(w_m;θ))/(p_k(w_m)+L_m(w_m;θ))^k+1×(1-w_m^k),

where we used the condition ∂ K_m/∂ v|_v=w_m=0
and the definitions (<ref>) and (<ref>). Owing to
Lemma <ref>(b), the right-hand side of (<ref>) is
positive and, therefore, the function m↦ K_m(w_m;θ) is
monotone increasing as long as w_m<1 and
m<m_0:=1/2(q-1). Likewise, every critical point
w=w^(i) from the original (finite) set W_m generates a
continuously differentiable branch m↦ w_m^(i) as a
function of the increasing variable m, and an argument similar to
(<ref>), now based on Lemma <ref>(a), yields that
the corresponding function m↦max_w∈ W_m K_m(w;θ)
is monotone decreasing up to m=m_0.

If for some m∈ (m,m_0) it occurs that
w_m=1 then, by continuity, ∂
K_m/∂ v|_v=1=0, which implies, as was
shown before (see (<ref>)), that
L_m(w_m;θ)=1 and therefore
K_m(w_m;θ)=1 is the maximum value
of the function v↦ K_m(v;θ). Moreover,
combining the monotonicity properties established above with the
hypothetical inequality (<ref>), this implies

    1=K_m(w_m;θ)>
    K_m(w_m;θ)>max_w∈ W_mK_m(w;θ)>max_w∈
    W_mK_m(w;θ),

that is,

    max_w∈ W_mK_m(w;θ)<1.

But this cannot be true, because there is
w>v_m where
L_m(w;θ)=1, so that
K_m(w;θ)=1 is another maximum and,
hence, w∈ W_m, thus
contradicting (<ref>).

This shows that we can exploit the monotonicity properties with
respect to variable m up to the final value m=m_0=1/2(q-1),
so that

    K_m(w_m;θ)<K_m_0(w_m_0;θ)

and also

    max_w∈ W_m_0K_m_0(w;θ)<max_w∈
    W_mK_m(w;θ).

Combining (<ref>) and (<ref>)
with (<ref>), it follows that

    max_w∈ W_m_0 K_m_0(w;θ)
    <K_m_0(w_m_0;θ).

But this is impossible, since m_0=m_0' and, by the scaling
relation (<ref>) of Lemma <ref>,
K_m_0(v;θ)≡ K_m_0(v^-1;θ) (see
(<ref>)), which implies that the maximum values of the
function v↦ K_m_0(v;θ) over v<1 and v>1 must be
the same, in contradiction with the inequality (<ref>).

Thus, the hypothesis (<ref>) is false, together with the
assumption w_m<1 under case (ii) (i.e., with L^*_m(θ)>1).
This completes the proof of Lemma <ref>.




 §.§ Properties of θ_m and α_m(θ)


For each m=1,…,q-2 and m'=q-1-m, we have

    θ_m=θ_m'.

Moreover, the functions θ↦α_m(θ)
see  satisfy the
symmetry relation

    α_m'(θ)≡α_m(θ),    θ>θ_m.


Like in Lemma <ref>(c), denote
v_m^*:=v_m(θ_m). Observe that v_m^* satisfies the
conjugation property

    v_m'^*=1/v_m^*,     m=1,…,q-2,

where m'=q-1-m. Indeed, computing the left-hand side of (<ref>)
for v=1/v_m^* and with m replaced by m', we get, due to
Lemma <ref>(c),

    m'∑_i=1^k-1i(1/v_m^*)^k-i-m∑_i=1^k-1
    i(1/v_m^*)^i-k=-(m∑_i=1^k-1
    i (v_m^*)^k-i-m'∑_i=1^k-1i(v_m^*)^i-k)=0,

whence (<ref>) follows due to the uniqueness of solution.

Now, using (<ref>) and the scaling property
(<ref>), we have

    L_m'(v_m'^*;θ_m)   =L_m'((v_m^*)^-1;θ_m)
       =(v_m^*)^-kL_m(v_m^*;θ_m)=0,

according to (<ref>), and by the uniqueness of solution to
the equation L_m'(v_m'(θ);θ)=0 (see
Lemma <ref>(b)), the equality (<ref>)
follows.

Finally, the identity (<ref>) is valid due to the
definition (<ref>) and formula (<ref>).



Let q≥5, and set m_0:=⌊1/2(q-1)⌋. Then
for m=1,…,m_0-1

    θ_m   <θ_m+1,
    α_m(θ)   >α_m+1(θ),
        θ>θ_m+1.


Treating m as a continuous parameter (see
Remark <ref>), differentiate the identity
(<ref>) to obtain

    L_m(v^*_m;θ_m)/ m   =∂
    L_m(v;θ_m)/∂ v|_v=v_m^*×
    v_m^*/ m
         +∂ L_m(v_m^*;θ)/∂θ|_θ=θ_m×θ_m/
    m+∂ L_m(v;θ)/∂
    m|_v=v_m^*, θ=θ_m≡ 0.

Using (<ref>) and (<ref>), the last identity is reduced
to

    p_k(v_m^*) θ_m/ m+1-(v_m^*)^k=0,

which yields

    θ_m/ m=(v_m^*)^k-1/p_k(v_m^*).

Recalling that v_m^*>1 for all m<1/2(q-1) (see
Lemma <ref>(c)), from (<ref>) it follows that
θ_m/ m>0 for m<1/2(q-1). For integer
m=1,…,m_0, this transcribes as the
inequality (<ref>).

Turning to the proof of (<ref>), for a given
θ≥θ_1 let m^*≥1 be the root of the equation
θ_m=θ. We shall prove a (stronger) continuous
version of the inequality (<ref>), namely, that the
function m↦α_m(θ) (defined for m≥ m^*) is
monotone decreasing. As before, denote by w_m=w_m(θ) the
point where the function v↦ K_m(v;θ) attains its
maximum value, and set K_m^*(θ):=K_m(w_m(θ);θ).
Differentiating the function m↦ K_m^*(θ), we obtain
(see (<ref>))

    ∂ K_m^*/∂ m   =(p_k(w_m)+1)^k(p_k(w_m)-(k-1)
    L_m(w_m;θ))/(p_k(w_m)+L_m(w_m;θ))^k+1×(1-w_m^k).

Now, owing to Lemmas <ref> and <ref> (see also
Remark <ref>), the right-hand side of (<ref>) is
negative for all m∈[m^*,1/2(q-1)) and,
therefore, the function m↦ K_m^*(θ) is monotone
decreasing in the closed interval [m^*,1/2(q-1)].
By the definition (<ref>), the same holds for the
function m↦α_m(θ), as claimed.



For all m=1,…,q-2, the functions
θ↦α_m(θ) defined by formula (<ref>)
satisfy the upper bound

    α_m(θ)<k-1,    θ>θ_m.

Moreover, they have the following “boundary” values,

    lim_θ↓θ_mα_m(θ)=-∞,    lim_θ↑∞α_m(θ)=k-1.



Let w_m=w_m(θ) be the point of maximum of the function
v↦ K_m(v;θ), so that K_m^*(θ)=K_m(w_m;θ).
Treating the term L_m=L_m(v;θ) in the expression (<ref>)
as an independent parameter L≥ 0, we can write

    K_m^*(θ)≤ (p_k(w_m)+1)^kmax_L≥0L/(p_k(w_m)+L)^k.

By differentiation, it easy to verify that the maximum on the
right-hand side of (<ref>) is attained at
L_0:=p_k(w_m)/(k-1), hence

    K_m^*(θ)   ≤ (p_k(w_m)+1)^kL/(p_k(w_m)+L)^k|_L=L_0
       =(k-1/k)^k(1+1/p_k(w_m))^kp_k(w_m)/k-1
       ≤p_k(w_m)/k-1.

Furthermore, w_m<θ by Lemma <ref>, so that

    p_m(w_m)<p_m(θ)<(k-1)θ^k-1.

Substituting this estimate into the right-hand side of
(<ref>), we obtain

    K_m^*(θ)< θ^k-1,

and therefore (see (<ref>))

    α_m(θ)=ln K_m^*(θ)/lnθ<k-1,

which proves the bound (<ref>). In particular, this
implies that

    lim sup_θ→∞α_m(θ)≤ k-1.


To obtain a matching lower bound, take a specific value

    v=v_0:=t/m(1-ln t/t),     t:=θ-1,

then, as t→∞,

    p_k(v_0)   =t^k-1/m^k-1(1-ln
    t/t)^k-1+O(t^k-2)
       =t^k-1/m^k-1(1-(k-1)ln
    t/t)+O(t^k-2)

and

    L_m(v_0;θ)   =t p_k(v_0)-m v_0^k-m'
       =t^k/m^k-1(1-(k-1)ln
    t/t)-t^k/m^k-1(1-kln
    t/t)+O(t^k-1)
       =t^k-1ln t/m^k-1+O(t^k-1).

Hence,

    p_k(v_0)+L_m(v_0;θ) ∼t^k-1ln t/m^k-1

and

    K_m(v_0;θ)   =L_m(v_0;θ)(p_k(v_0)+1)^k/(p_k(v_0)+L_m(v_0;θ))^k∼(t/mln t)^k-1.

Therefore,

    ln K_m(v_0;θ)∼ (k-1)ln t∼ (k-1)lnθ,    θ→∞,

so that

    lim inf_θ→∞α_m(θ) ≥lim_θ→∞ln K_m(v_0;θ)/lnθ=k-1.

Thus, combining (<ref>) and (<ref>), we
obtain the second limit in (<ref>).

Finally, we turn to the proof of the first limit
in (<ref>). By virtue of
Proposition <ref>, we may assume that
m≤1/2(q-1). Then, by Lemma <ref>, w_m≥1 and
therefore p_k(w_m)≥ k-1. Hence, from the definition (<ref>)
we get

    0< K_m^*(θ)   ≤
    L_m(w_m;θ)(1+1/p_k(w_m))^k
       ≤ L_m^*(θ)(k/k-1)^k,

where
L_m^*(θ)=L_m(v_m(θ);θ)=max_v>0L_m(v;θ). By
continuity,

    lim_θ↓θ_m L_m^*(θ)= L_m^*(θ_m)=0,

and it follows from the bound (<ref>) that
lim_θ↓θ_m K_m^*(θ)=0, which implies the
first limit in (<ref>). Thus, the proof of
Proposition <ref> is complete.



For each m=1,…,q-2, the function θ↦ K_m^*(θ)
is monotone increasing for θ>θ_m.

By virtue of Lemma <ref>,
K_m^*(θ)=K_m'^*(θ), where m'=q-1-m; hence, it
suffices to prove the claim for m in the range 1≤ m≤1/2(q-1). Using the definitions (<ref>) and (<ref>),
differentiate with respect to θ to obtain

    K_m^*/θ   =∂ K_m/∂
    v|_v=w_m× w_m/θ
    +∂ K_m/∂
    L|_v=w_m, L=L_m(w_m;θ)×∂
    L_m/∂θ|_v=w_m
       =(p_k(w_m)+1)^k(p_k(w_m)-(k-1)
    L_m(w_m;θ))/(p_k(w_m)+L_m(w_m;θ))^k+1×
    p_k(w_m),

on account of the identity ∂ K_m/∂
v|_v=w_m≡ 0. To complete the proof, it remains to notice
that the right-hand side of (<ref>) is positive due to
Lemma <ref>(b).



The result of Proposition <ref> is not trivial (unlike
the similar statement for L^*_m(θ), see the footnote in the
proof of Lemma <ref>(b)), because, for each v>0, we
have that L_m(v;θ)→∞ and, therefore, K_m(v;θ)→
0 as θ→∞ (see formula (<ref>)).





For each m=1,…,q-2, the function θ↦α_m(θ)=ln K_m^*(θ)/lnθ is monotone
increasing.

This conjecture is confirmed by computer plots (see
Figure <ref>) and is easy to prove at least for θ≤θ_m^0, where θ_m^0 is the root of the
equation K_m^*(θ)=1 (cf.Proposition <ref>(a)); that is, α_m(θ)≤
0 for θ≤θ_m^0. Indeed, for any
θ∈(θ_m,θ_m^0], we have

    α_m(θ)/θ   =
    K_m^*(θ)/θ×1/lnθ·
    K_m^*(θ)-ln K_m^*(θ)/θln^2θ>0,

because K_m^*/θ>0 (Proposition <ref>),
whereas ln K_m^*(θ)≤ln K_m^*(θ_m^0)=0.



 §.§ Zeros of α_±(θ) and α_m(θ)

Recall that the functions α_±(θ) and
α_m(θ) are defined in (<ref>) and
(<ref>), respectively. As was observed in numerical
examples (see Figure <ref> and also Figures <ref>
and <ref>), the functions α_-(θ) and
α_1(θ) have the same zero, θ_0^-=θ_1^0,
whereas α_+(θ)=0 at θ^+_0=1+q/(k-1). In this
subsection, we give a proof of these observations.

Let us first state and prove a lemma. Recall the notation
L_m^*(θ)=max_v∈𝒱_m^+(θ)L_m(v;θ) and
K_m^*(θ)=max_v∈𝒱_m^+(θ)K_m(v;θ).

Let q≥3 and k≥2.


  (a)
For any m∈[1,q-2], if L_m(v;θ)<1 for some v≥1 and
θ>θ_m then K_m(v;θ)<1.


  (b) Let 1≤ m≤1/2(q-1). If L_m^*(θ)<1 for some
θ>θ_m then K_m^*(θ)<1. In particular,
L_m^*(θ)=1 if and only if K_m^*(θ)=1.


(a) Denoting s:= (L_m(v;θ))^1/k<1 and using the
definition (<ref>), the required inequality K_m(v;θ)<1
can be rewritten as

    p_k(v)> s-s^k/1-s=p_k(s),

and the last inequality holds by monotonicity of p_k(v), since
v≥ 1>s.

(b) Let v=w_m be such that K_m(w_m;θ)=K_m^*(θ)<1; by
Lemma <ref>, w_m≥ 1. On the other hand,
L_m(w_m;θ)≤ L_m^*(θ)<1, and by part (a) it follows
that K_m(w_m;θ)<1. The last claim in part (b) then follows by
continuity and monotonicity of both L_m^*(θ) and
K_m^*(θ) (see Lemma <ref>(b) and
Proposition <ref>, respectively), also recalling that
L_m(v;θ)=1 implies K_m(v;θ)=1 (see (<ref>)).


 Let q≥ 3, and set
m_0:=⌊12(q-1)⌋.


  (a)
For each m in the range 1≤ m≤ m_0, the function
α_m(θ) has a unique zero given by

    θ_m^0=m(v_m^0)^k+m'+1/p_k(v_m^0),

where v_m^0>1 is a sole positive root of the equation

    m∑_i=1^k-1i v^k-i-(m'+1)∑_i=1^k-1i
    v^i-k=0.



  (b)

  (i)
The function α_+(θ) has a unique zero given by
θ^+_0=1+q/k-1. Moreover,
α'_+(θ^+_0)=0.


  (ii)
The function α_-(θ) has a unique zero
θ^-_0, which coincides with the zero θ_1^0
of the function α_1(θ).



  (c) The zeros
θ_1^0,…,θ_m_0^0 follow in ascending
order and are strictly below θ_0^+,

    θ_1^0<…<θ_m_0^0<θ_0^+=1+q/k-1.




(a) By the definition (<ref>), the condition
α_m(θ)=0 means that K_m^*(θ)=1 and hence, by
Lemma <ref>(b), L_m^*(θ)=1. Eliminating θ
from the system of equations L_m(v;θ)=1, ∂ L_m(v;θ)/∂ v=0 gives for the root v=v_m^0 the
equation (cf. (<ref>))

    mk v^k-1p_k(v)-(m v^k+m'+1) p'_k(v)=0,

which can be rearranged to the form (<ref>). Uniqueness of
positive solution v_m^0 of the equation (<ref>) is
obvious, noting that the left-hand side of (<ref>) is a
continuous, increasing function in v>0, with the range from
-∞ to +∞. To show that v_m^0>1, it suffices to
check that the left-hand side of (<ref>) at v=1 is negative,
which is indeed true since 2m≤ q-1<q. Expressing θ from
the equation L_m(v^0_m;θ)=1, we obtain
formula (<ref>).

(b) In the limit α→0, the equation (<ref>) always has
root u=1, while for u1, by virtue of the identity
(<ref>), it is reduced to equation (<ref>) with m=1.
Using the notation (<ref>), the latter equation can be
rewritten as L_1(u;θ)=1, which in turn has up to two
(positive) roots (see Lemma <ref>). In total, there are
three positive roots, and for this number to reduce to two
(which is the condition of belonging to the curves
y=α_±(θ)), either (i) one zero of the function
u↦ L_1(u;θ)-1 must coincide with u=1, or (ii) the
equation L_1(u;θ)=1 must have a double root, thus also
satisfying the condition ∂ L_1(u;θ)/∂ u =0.

In case (i), the condition L_1(u;θ)|_u=1=1 transcribes
as (θ-1)(k-1)-(q-1)=1, which immediately yields the root
θ_0^+=1+q/(k-1). According to the substitution (<ref>)
(with α=0), the corresponding root of the quadratic equation
(<ref>) is given by x=θ_0^+/(q-1), which
appears to be the smaller of the two roots, x=x_-.
Therefore, in view of formulas (<ref>) and
(<ref>), the value θ_0^+ is a zero of the
function  α_+(θ). Indeed, using the definition
(<ref>) of b=b(θ), the second root of
(<ref>) is found to be

    x_+=b/x_-=θ_0^+(θ_0^+
    +q-2)/q-1·q-1/θ_0^+=θ_0^++q-2
    >θ_0^+/q-1=x_-,

as claimed.

In case (ii), according to the proof of part (a), the unique
solution of the system L_1(u;θ)=1, ∂
L_1(u;θ)/∂ u=0 is given by
(u,θ)=(v_1^0,θ_1^0), where

    θ_1^0=1+(v_1^0)^k+2/p_k(v_1^0)

and v_1^0>1 is a sole root of the equation (<ref>) with m=1,
that is,

    ∑_i=1^k-1i v^k-i-(q-1)∑_i=1^k-1i v^i-k=0.

Again by the substitution (<ref>) with α=0, the
corresponding root of the quadratic equation
(<ref>) is given by

    x=θ_1^0 (v_1^0)^k /q-1,
 and we wish to
prove that this is the bigger of the two roots, x=x_+,
which would imply like before that θ_1^ 0 is a zero of
the function α_-(θ). Since the other root of
(<ref>) equals b/x, with b=b(θ) defined
in (<ref>), our claim is expressed as x>b/x, that is,

    (v_1^0)^2k>(q-1)(1+q-2/θ_1^0).

Furthermore, recalling that θ_1^0>1
(see (<ref>)), we have

    1+q-2/θ_1^0>q-1,

so for the proof of (<ref>) it suffices to show that

    v_1^0≥(q-1)^1/k.

Since the function on the left-hand side of (<ref>) is monotone
increasing, we only need to check that its value at v=(q-1)^1/k
is non-positive, that is,

    ∑_i=1^k-1i v^k-i-∑_i=1^k-1i v^i-k≤ 0.

which can be rewritten as

    ∑_i=1^k-1(2 i-k) v^i-k/2≥0.

Now, the latter inequality holds because the left-hand side is
obviously monotone increasing as a function of v≥ 1, being equal
to 0 at v=1. Hence, (<ref>) follows.

Thus, we have proved that θ_0^+=1+q/(k-1) and
θ_0^-=θ_1^0 are the sole roots of the functions
α_+(θ) and α_-(θ), respectively.

Finally, since the function α_+(θ) is known to be
positive both at θ_ c<θ_0^+ and at infinity (see
Proposition <ref>), it readily follows that it
has a minimum value 0 at θ=θ_0^+, hence
α'_+(θ)=0, as claimed in part (b)(i).

(c) The ordering inequalities between (θ_m^0) in
(<ref>) readily follow from the monotonicity property
(<ref>) proved in Proposition <ref>.
Thus, it remains to show that θ_m^0<θ_0^+. Observe
that the value L_m(v;θ)|_v=1=(θ-1)(k-1)-(q-1)
does not depend on m. Recalling that L_1(1;θ_0^+)=1 (see
the proof of part (a)), we get that L_m(1;θ_0^+)=1, but

    .∂ L_m (v;θ_0^+/∂
    v|_v=1   =(θ_0^+-1)
    p_k'(1)-mk
       =q/k-1·k(k-1)/2-mk
       =k(q/2-m)>0,

since m≤1/2(q-1)<q/2. Hence, L_m^*(θ_0^+)>L_m
(1;θ_0^+)=1, and by monotonicity of the function
θ↦ L_m(v;θ) it follows, according to the proof in
part (a), that θ_m^0<θ_0^+.

An alternative simple argument is that, as shown in part (a), the
maximum value of v↦ L_m(v;θ_m^0)=1 is attained at
v=v_m^0>1, hence L_m(1;θ_m^0)<1 and, again by
monotonicity, it follows that θ_0^+>θ_m^0.


A version of Proposition <ref>(b) for q=2 is easy
to obtain.

In the case q=2, the functions α_±(θ) have a
unique zero at θ_ c=k+1/k-1, which coincides
with θ_0^+=1+2/(k-1). Moreover, α'_±(θ_
c+)=0.

By Lemma <ref> with q=2, we have
α_-(θ)=-α_+(θ) for all θ≥θ_ c=k+1/k-1. Thus, it suffices to consider the
function α_+(θ). Treating the index q≥2 as a
continuous variable and taking the limit from the domain q>2 as
q→ 2+, we see that the unique zero of α_+(θ),
given by θ_0^+(k,q)=1+q/(k-1), converges to
1+2/(k-1)=k+1/k-1=θ_ c. On the other hand, the
derivative α'_+ vanishes at θ_0^+(k,q) for each
q>2, hence its limiting value at θ_ c is also zero,
that is, α'_+(θ_ c+)=0, as claimed.

This result can also be obtained by a direct calculation. Namely,
for q=2 the definition (<ref>) is reduced to
b(θ)=θ^2. From equation (<ref>) with
q=2, it is easy to see that x_±(θ)→θ_ c as
θ→θ_ c+. Moreover, a simple asymptotic analysis
shows that

    x_±(θ)=θ_ c±√(2 kθ_
    c(θ-θ_ c))+(k+1)(θ-θ_
    c)+o(θ-θ_ c),    θ→θ_ c+.

Hence, from (<ref>) it follows

    ln a_±(θ)=-(k+1)lnθ_
    c-(k-1)(θ-θ_ c)+o(θ-θ_ c),    θ→θ_ c+.

Finally, substituting this into (<ref>) we obtain

    α_±(θ)   =-(k+1)+(k+1)lnθ_ c+(k-1)(θ-θ_ c)+o(θ-θ_ c)/lnθ_ c+θ_ c^-1
    (θ-θ_ c)+o(θ-θ_ c)
       =θ-θ_ c/lnθ_ c((k-1)-k+1/θ_ c)+o(θ-θ_ c)
       =o(θ-θ_ c),    θ→θ_ c+,

which implies that α'±-(θ_ c+)=0, as claimed.



Consider the case k=2. Then the zero of the function
α_+(θ)=0 specializes to θ^+_0=1+q.
Furthermore, the equation (<ref>) is easily solved to yield
v_m^0=√((q-m)/m), and from (<ref>) we readily find

    θ_m^0=1+2√(m(q-m)),     1≤ m≤12(q-1).

It is of interest to note, by comparing (<ref>) with
(<ref>), that θ_m^0=θ_m+1 (cf.<cit.>). Finally, taking m=1 in
(<ref>) gives
θ^-_0=θ_1^0=1+2√(q-1).





 §.§ Acknowledgements


This research was partially supported by a Royal Society
International Exchange Grant IES\R2\181078 (December 2018 – March 2019). U.A.R. acknowledges support
by the Institute for Advanced Study of Aix-Marseille University
(IMéRA) under a research residency scheme (February–July 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, February–March
2016). Research of L.V.B. was partially supported by a visiting
grant from SFB 701 (Bielefeld University, June 2015).





§ PROOF OF PROPOSITION <REF>


Necessity. Assume that the measure μ^h is translation
invariant, that is, the condition (<ref>) holds. Pick the set
⊂ V to be the unit ball V_1≡
V_1(x_∘)={x_∘, x_1,…,x_k+1} centred at the root
x_∘, with ∂{x_∘}={x_1,…,x_k+1}, where
the numbering in (x_j) is consistent with the bijection
𝔟 V→𝒜_k, that is,
x_j=𝔟^-1(a_j) (see Section <ref>). For
z∈ V, the shifted set
V_1(z):=T_z(V_1)={z,z_1,…,z_k+1}
(where z_j=T_z(x_j), j=1,…,k+1) is the
unit ball centred at z=T_z(x_∘), so that
∂{z}={z_1,…,z_k+1}. Consider an arbitrary
configuration ς∈^V_1, with spin values

    ς(x_∘)=i_0,    ς(x_j)=i_j,     
    j=1,…,k+1.

Note that if y=T_z(x) then, according to
(<ref>),

    (T_zς)(y)=ς(T^-1_z(y))=ς(x).

Hence, the property (<ref>) of translation invariance of μ^h
specializes as follows,

    μ^h(σ∈^Vσ(z)=i_0, σ(z_1)=i_1,…, σ(z_k+1)=i_k+1)
    
    =μ^h(σ∈^Vσ(x_∘)=i_0, σ(x_1)=i_1,…, σ(x_k+1)=i_k+1).


Using formulas (<ref>) and (<ref>),
and cancelling the common term β∑_j=1^k+1
Jδ_i_0,i_j, the equality (<ref>) is reduced to

    ∑_j=1^k+1[ξ_i_j(z_j)+h^†_i_j(z_j,z)]+ξ_i_0(z)=∑_j=1^k+1[ξ_i_j(x_j)+h^†_i_j(x_j,x_∘)]
    +ξ_i_0(x_∘)+1/βlnZ_V_1(z)/Z_V_1(x_∘).

Varying i_0∈ in (<ref>) (whilst keeping all other
i_j fixed) shows that the difference
ξ_i_0(z)-ξ_i_0(x_∘) does not depend on i_0. More
precisely, on subtracting from (<ref>) the same equality with
i_0=q, we obtain

    ξ_i(z)-ξ_q(z)=ξ_i(x_∘)-ξ_q(x_∘),    
    i=1,…,q-1.

Since (<ref>) holds for any z∈ V, this
proves (<ref>) in view of the notation (<ref>).

Furthermore, by virtue of (<ref>) the equality (<ref>)
becomes

    ∑_j=1^k+1 h^†_i_j(z_j,z)=∑_j=1^k+1
    h^†_i_j(x_j,x_∘)+∑_x∈
    V_1(x_∘)ξ_q(x)-∑_y∈
    V_1(z)ξ_q(y)+1/βlnZ_V_1(z)/Z_V_1(x_∘).

Similarly, varying the values i_1,…,i_k+1 in (<ref>)
(one at a time) yields, for each j=1,…,k+1,

    ȟ^†_i(z_j,z)=ȟ^†_i(x_j,x_∘),    
    i=1,…,q-1.

On the other hand, fix j∈{1,…,k+1} and consider the shift
T_z_j, resulting in

    T_z_j(x_∘)=z_j,    T_z_j(x_j)=z.

The latter equality follows by recalling the definition of conjugate
translations (see (<ref>)) and noting that
z_j=T_z(x_j)=𝔟^-1(𝔟(z)𝔟(x_j))
=𝔟^-1(𝔟(z) a_j) and, therefore,

    T_z_j(x_j)=𝔟^-1(𝔟(z_j)𝔟(x_j))=𝔟^-1(𝔟(z)
    a_j^2)=𝔟^-1(𝔟(z))=z,

because a_j^2=e (see Section <ref>). Hence, for
T_z_j the result (<ref>) transforms into

    ȟ^†_i(z,z_j)=ȟ^†_i(x_j,x_∘),    
    i=1,…,q-1.

Comparing (<ref>) and (<ref>), we conclude that
ȟ^†_i(z_j,z)=ȟ^†_i(z,z_j), and the claim
(<ref>) follows. Finally, let y=T_v(z) and
y_j=T_v(z_j), for some v∈ V. Observe that

    y=T_y(x_∘),     y_j=T_y(x_j).

The first equality in (<ref>) is automatic; to check the
second one, note that

𝔟(y_j)   =𝔟(v)𝔟(z_j)           [y_j=T_v(z_j)]

   =𝔟(v)( 𝔟(z)𝔟(x_j))      
[z_j=T_z(x_j)]

   =(𝔟(v)𝔟(z))𝔟(x_j)

   =𝔟(y)𝔟(x_j)       [y=T_v(z)].

That is, 𝔟(y_j)=𝔟(y)𝔟(x_j)
and (<ref>) follows. Thus, formula (<ref>) applied
to the edge ⟨ y_j,y⟩ gives

    ȟ^†_i(y_j,y)=ȟ^†_i(x_j,x_∘),    
    i=1,…,q-1.

Combined with (<ref>), this implies

    ȟ^†_i(z_j,z)=ȟ^†_i(y_j,y)=ȟ^†_i(T_v(z_j),T_v(z)),

and (<ref>) follows. This completes the “only if” part of
the proof.

Sufficiency. Suppose that the conditions (<ref>),
(<ref>) and (<ref>) are satisfied. It suffices to
verify formula (<ref>) for the balls V_n (n≥ 1). For z∈
V, denote :=T_z(V_n-1), then
=T_z(V_n) and
∂=T_z(W_n). Furthermore, for
ς∈^V_n set
ς_z:=T_z(ς)∈^
(see (<ref>)). Observe that if y=T_z(x)
then, according to (<ref>),

    ς_z(y)=ς(T^-1_z(y))=ς(x).

Hence, recalling (<ref>), we have

    H_(ς_z)    =-∑_⟨ y,
    y'⟩∈ E_J
    δ_ς_z(y),ς_z(y')-∑_y∈ξ_ς_z(y)(y)
       =-∑_⟨ x, x'⟩∈ E_nJ
    δ_ς(x),ς(x')-∑_x∈ V_nξ_ς(x)(T_z(x))
       =H_n(ς)+∑_x∈
    V_n(ξ_q(x)-ξ_q(T_z(x))),

where at the last step we used the property (<ref>). Thus,
from formula (<ref>) we obtain, omitting factors
not depending on ς,

    μ^h(σ_=ς_ z)∝μ^h(σ_V_n=ς)·exp{β∑_y∈∂h^†_ς_
    z(y)(y,y_)-β∑_x∈
    W_nh_ς(x)(x)},

where y_ is the unique neighbour of
y∈∂ in . Note that if
y=T_z(x), with x∈∂ V_n-1=W_n, then
y_=T_z(x'), where x'∈ W_n-1
is the unique vertex


such that x∈ S(x'). Thus, using (<ref>), (<ref>)
and (<ref>), we can write

    exp{β∑_y∈∂h^†_ς_
    z(y)(y,y_)}   =
    exp{β∑_x∈
    W_nh^†_ς(x)(T_z(x),T_z(x'))}
       ∝exp{β∑_x∈
    W_nh^†_ς(x)(x,x')}
       ∝exp{β∑_x∈
    W_nh_ς(x)(x)}.

Returning to (<ref>), this gives
μ^h(σ_=ς_z)∝μ^h(σ_V_n=ς), and since

    ∑_ς∈^V_nμ^h(σ_=ς_z)=1
    =∑_ς∈^V_nμ^h(σ_V_n=ς),

it follows that
μ^h(σ_=ς_z)=
μ^h(σ_V_n=ς), and the proof of the “if” part
is complete.



§ PROOF OF LEMMA <REF>



 §.§ Proof of part (a)

Denote t:=θ-1. For q=3 (i.e., with m=m'=1), the
partial derivative ∂ K_1/∂ v (see (<ref>)) can
be represented as

    ∂ K_1/∂ v   =
    (p_k+1)^k-1/(p_k+L_m)^k+1 {(p_k+1+(1-L_1)(k-1))L'_1(p_k-(k-1))
            - (1-L_1) k  L_1 (p_k'-k
    (k-1)/2)+k(k-1)/2(1-L_1)(2L_1'-k
    L_1)},

where L_1=t p_k-v^k-1 and L_1'=∂ L_1/∂ v=t
p'_k-k v^k-1.

Note that ∂ K_1/∂ v|_v=1=0
(cf. (<ref>)); this follows without calculations from the
scaling property K_1(v;t)=K_1(v^-1;t) (see (<ref>)).
More explicitly, using the formulas

    p_k|_v=1=k-1,      p'_k|_v=1=k
    (k-1)/2,

it is easy to see that the terms in (<ref>) vanish at v=1.
We will also need the formula

    p”_k|_v=1=k (k-1)(k-2)/3.

Such expressions can be obtained by successively differentiating (at
v=1) the identity

    v^k-1≡(v-1)(p_k+1).


To compute the second-order derivative ∂^2 K_1/∂ v^2
at v=1, we need to differentiate in (<ref>) only the factors
that vanish at v=1, setting v=1 elsewhere. Hence,

    .∂^2 K_1/∂
    v^2|_v=1   =(p_k+1)^k-1/(p_k+L_1)^k+1 {(p_k+1)
    L'_1 p'_k
          +(1-L_1)[(k-1) L'_1 p'_k- k  L_1 p_k”
    +k(k-1)/2(2L”_1-k
    L'_1)]}|_v=1.
 Using
(<ref>) and (<ref>), we find

    L_1|_v=1   =t p_k(1)-2=t(k-1)-2,
    L_1'|_v=1   =t p'_k(1)-k=tk(k-1)/2 -k,
    L_1”|_v=1   =t p”_k(1)-k(k-1)=tk(k-1)(k-2)/3 -k(k-1).

Finally, substituting formulas (<ref>), (<ref>) and
(<ref>)–(<ref>) into (<ref>), after simple
manipulations (verified with Maple) we obtain

    .∂^2 K_1/∂ v^2|_v=1   =(k/(k-1)(t+1)-2)^k+1 k-1/2
           ×((k-1)^2/2 t^2+(k-1)(7k-11)/6 t-3k+1).
 The quadratic polynomial in (<ref>) has one positive
zero t=t^*,

    t^*=11-7k+√(49k^2+62k+49)/6(k-1),

which corresponds to θ̃_1=t^*+1 as defined
in (<ref>). Hence, ∂^2 K_1/∂
v^2|_v=1>0 for θ>θ̃_1, which implies that
v=1 is a local minimum of the function v↦ K_1(v;θ).
This completes the proof of Lemma <ref>(a).



 §.§ Proof of part (b)

It suffices to show, for 1≤θ≤θ̃_1 (i.e.,
0≤ t≤ t^*), that v=1 is the sole root of the equation
∂ K_1/∂ v=0. A plausible general scheme of the proof
of the latter statement may be as follows.



  (1)
First, the condition ∂ K_1/∂ v=0 (see (<ref>))
is reduced to P(v;t)=0, where P(v;t) is a polynomial in v (and
also a quadratic polynomial in t).


  (2)
Since P(1;t)=0, the quotient R(v;t)=P(v;t)/(v-1) is a polynomial
in v, and we wish to prove that R(v;t)<0 for any t≤ t^*
and all v1.


  (3) According to the proof
in Section <ref>, the condition ∂^2K_1/∂
v^2|_v=1≤ 0 is satisfied whenever R(1;t)≤ 0; moreover, the
critical value t^* is determined by the quadratic equation
R(1;t^*)=0, which implies that R(1;t)≤0 for t∈[0,t^*].


  (4) Bearing in mind the invariance of K_1(v;t)
under the map v↦ v^-1, it should be possible to represent
the polynomial R(v;t) in the form

    R(v;t)=χ(v)·R(y;t),     y=v+v^-1≥2,

where χ(v)>0 and R(y;t) is a polynomial in y.


  (5)
The crucial step is to show that, for each t∈[0,t^*], the
function y↦R(y;t) is monotone
decreasing.


  (6) Finally, using steps (3) to (5), for any y>2 (i.e., v1) we have

    R(y;t)<R(2;t)= R(1;t)/χ(1)≤
    0,     0≤ t≤ t^*.

Hence, for all t∈[0,t^*] and v1, we get

    R(v;t)=χ(v)·R(y;t)<0,

as required.


In what follows, we implement this scheme in more detail for the
cases k=2,3,4. All calculations were done analytically and also
verified using Maple.



  §.§.§ Case k=2
 According to formula (<ref>),
t^*=1/2(√(41)-1). The polynomial P(v;t) is
found to be

    P(v;t)   =t^2v^2-t^2v-tv^3+4tv^2-4tv+t-4v^3+2v^2-2v+4
       =(v-1){t^2 v-t(v^2-3v+1)-(4v^2+2v+4)}.

Hence,

    R(v;t)   =t^2 v-t(v^2-3v+1)-(4v^2+2v+4
       =v(t^2-t(v-3+1/v)-4(v+1/v)-2)
       =v(t^2-t(y-3)-4y-2),

where y=v+v^-1. This gives

    R(y;t)=t^2-t(y-3)-4y-2=-y(t+4)+t^2+3t-2,

which is clearly a decreasing function of y for any t≥0.



  §.§.§ Case k=3
 By (<ref>), we have t^*=4/3.
The quotient R(v;t)=P(v;t)/(v-1) is explicitly given by

    R(t,v)   =(v+1)(t+3)(2tv^3+2tv -2v^4 -2+ 5tv^2 -4v^3-4v)
       =v^2(v+1)(t+3)(2t(v+1/v)-2(v^2+1/v^2)+5t-4(v+1/v))
       =v^2(v+1)(t+3)(2ty-2(y^2-2)+5t-4y).

Thus,

    R(y;t)=(t+3)(2ty-2(y^2-2)+5t-4y),

and it is easy to check that this function is decreasing in y for
any t≤ t^*=4/3.



  §.§.§ Case  k=4
 Elementary but tedious
calculations yield

    R(v;t)   =t^2(3v^7+11v^6+25v^5+30v^4+25v^3+11v^2+3v)
         
    -t(3v^8+v^7-13v^6-57v^5-72v^4-57v^3-13v^2+v+3)
           
    -(8v^8+24v^7+48v^6+36v^5+32v^4+36v^3+48v^2+24v+8).

Rearranging under the substitution y=v+v^-1 gives
R(v;t)=v^4R(y;t) with

    R(y;t)   =t^2(3y^3+11y^2+16y+8)
         -t(3y^4+y^3-25y^2-60y-40)
           -(8y^4+24y^3+16y^2-36y-48).
 In
particular, if y=2 then

    R(2;t)=12(9 t^2+17 t-22)=0

for t=1/18(√(1081)-17)=t^*
(cf. (<ref>)), as it should be.

Finally, we need to verify that the function y↦R(y;t) is decreasing for any t∈[0,t^*].
Unfortunately (but inevitably), technicalities involved in a purely
analytic check become quite substantial; however, using Maple
to plot the graph of (<ref>), with parameter t ranging
from 0 to t^*, makes the monotonicity evident.


99


Abou-Chacra
Abou-Chacra, R., Anderson, P.W. and Thouless, D.J. A selfconsistent
theory  of localization. J. Phys. C 6 (1973),
1734–1752.
(https://doi.org/10.1088/0022-3719/6/10/009doi:10.1088/0022-3719/6/10/009)

Aizenman
Aizenman, M. and Warzel, S. The canopy graph and level statistics
for random operators on trees. Math. Phys. Anal. Geom.
9 (2007), 291–333.
(https://doi.org/10.1007/s11040-007-9018-3doi:10.1007/s11040-007-9018-3, 2329431)






Athreya
Athreya, S., Bandyopadhyay, A. and Dasgupta, A. Random walks in
i.i.d. random environment on Cayley trees. Statist. Probab.Lett. 92 (2014), 39–44.
(https://doi.org/10.1016/j.spl.2014.04.026doi:10.1016/j.spl.2014.04.026, 3230470)


Ba Baxter, R.J. Exactly Solved Models in Statistical
Mechanics. Academic Press, London, 1982. (0690578)







Be1
Beaudin, L.,  Ellis-Monaghan, J., Pangborn, G. and Shrock, R. A
little statistical mechanics for the graph theorist. Discrete
Math. 310 (2010),

2037–2053.
(https://doi.org/10.1016/j.disc.2010.03.011doi:10.1016/j.disc.2010.03.011, 2629922)

Bethe
Bethe, H.A. Statistical theory of superlattices. Proc. Roy.Soc. A 150 (1935), 552–575.
(https://doi.org/10.1098/rspa.1935.0122doi:10.1098/rspa.1935.0122)

BRSSZ Bleher, P., Ruiz, J., Schonmann, R.H.,
Shlosman, S. and Zagrebnov, V. Rigidity of the critical phases on a
Cayley tree. Mosc. Math. J. 1 (2001),

345–363.
(https://doi.org/10.17323/1609-4514-2001-1-3-345-363doi:10.17323/1609-4514-2001-1-3-345-363,
1877597)


BRZ Bleher, P.M., Ruiz, J. and Zagrebnov, V.A. On the
phase diagram of the random field Ising model on the Bethe lattice.
J. Stat. Phys. 93 (1998), 33–78.
(https://doi.org/10.1023/B:JOSS.0000026727.43077.49doi:10.1023/B:JOSS.0000026727.43077.49, 1656364)


Bov
Bovier, A. Statistical Mechanics of Disordered Systems: A
Mathematical Perspective. Cambridge Series in Statistical and
Probabilistic Mathematics, 18. Cambridge University Press,
Cambridge, 2006.
(https://doi.org/10.1017/CBO9780511616808doi:10.1017/CBO9780511616808, 2252929)

Boy Boykov, Y.,  Veksler, O. and Zabih R. Fast approximate energy minimization via graph cuts.
IEEE Trans. Pattern Anal. Machine Intell. 23

(2001), 1222–1239.
(https://doi.org/10.1109/34.969114doi:10.1109/34.969114)

Bricmont
Bricmont, J., Kuroda, K. and Lebowitz, J.L. First order phase
transitions in lattice and continuous systems: Extension of
Pirogov–Sinai theory. Comm. Math. Phys. 101 (1985),

501–538. (https://doi.org/10.1007/BF01210743doi:10.1007/BF01210743, 0815198)

Br
Brookings, T., Carlson, J.M. and Doyle, J. Three mechanisms for
power laws on the Cayley tree. Phys. Rev. E 72 (2005),
056120, 18 pp.
(https://doi.org/10.1103/PhysRevE.72.056120doi:10.1103/PhysRevE.72.056120, 2198305)


Chen
Chen, M.-S., Onsager, L., Bonner, J. and Nagle, J. Hopping of ions
in ice. J. Chem. Phys. 60 (1974), 405–419.
(https://doi.org/10.1063/1.168105660doi:10.1063/1.168105660)

DMR
Daskalakis, C., Mossel, E. and Roch, S. Evolutionary trees and the
Ising model on the Bethe lattice: a proof of Steel's conjecture.
Probab. Theory Related Fields 149 (2011),

149–189.
(https://doi.org/10.1007/s00440-009-0246-2doi:10.1007/s00440-009-0246-2, 2773028)


Dembo2
Dembo, A., Montanari, A., Sly, A. and Sun, N. The replica symmetric
solution for Potts models on d-regular graphs. Comm. Math.Phys. 327 (2014), 551–575.
(https://doi.org/10.1007/s00220-014-1956-6doi:10.1007/s00220-014-1956-6, 3183409)








Domb
Domb, C. Configurational studies of the Potts models.
J. Phys. A

7 (1974), 1335–1348.
(https://doi.org/10.1088/0305-4470/7/11/013doi:10.1088/0305-4470/7/11/013)

Fr
Friedrich, F., Kempe, A., Liebscher, V. and Winkler, G.  Complexity
penalized M-estimation: fast computation. J. Comput.Graph. Statist. 17 (2008),
 
201–224.
(https://doi.org/10.1198/106186008X285591doi:10.1198/106186008X285591,
 2424802)

Galanis
Galanis, A., Štefankovič, D., Vigoda, E. and Yang, L.
Ferromagnetic Potts model: refined #BIS-hardness and related
results. SIAM J. Comput. 45 (2016),

2004–2065.
(https://doi.org/10.1137/140997580doi:10.1137/140997580, 3572375)


GRR
Gandolfo, D., Rahmatullaev, M.M. and Rozikov, U.A. Boundary
conditions for translation-invariant Gibbs measures of the Potts
model on Cayley trees.  J. Stat. Phys. 167 (2017),

1164–1179.
(https://doi.org/10.1007/s10955-017-1771-5doi:10.1007/s10955-017-1771-5, 3647056)










Ga0
Ganikhodjaev, N.N. Group presentations and automorphisms of the
Cayley tree. (Russian) Dokl. Akad. Nauk Respub.Uzbekistan, no. 4 (1994), 3–5.

GaRo Ganikhodjaev, N. and Rozikov, U. On disordered
phase in the ferromagnetic Potts model on the Bethe lattice.
Osaka J. Math. 37

(2000), 373–383.
(http://projecteuclid.org/euclid.ojm/1200789203http://projecteuclid.org/euclid.ojm/1200789203, 1772838)


Ge
Georgii, H.-O. Gibbs Measures and Phase Transitions, 2nd ed.
De Gruyter Studies in Mathematics, 9.

De Gruyter, Berlin, 2011. (2807681)

Green Green, P.J. and Richardson, S. Hidden Markov
models and disease mapping. J. Amer. Statist. Assoc. 97 (2002),

1055–1070.
(https://doi.org/10.1198/016214502388618870doi:10.1198/016214502388618870, 1951259)

Graner
Graner, F. and Glazier, J. Simulation of biological cell sorting
using a two-dimensional extended Potts model. Phys. Rev.Lett. 69 (1992),

2013–2017.
(https://doi.org/10.1103/PhysRevLett.69.2013doi:10.1103/PhysRevLett.69.2013)









Gr1
Grimmett, G. The Random-Cluster Model. Grundlehren der
mathematischen Wissenschaften [A Series of Comprehensive Studies in
Mathematics], 333. Springer, Berlin, 2006.
(https://doi.org/10.1007/978-3-540-32891-9doi:10.1007/978-3-540-32891-9, 2243761)

Hagg
Häggström, O. The random-cluster model on a
homogeneous tree. Probab. Theory Related Fields 104
(1996), 231–253.

(https://doi.org/10.1007/BF01247839doi:10.1007/BF01247839, 1373377)

Kotecky-Shlosman
Kotecký, R. and Shlosman, S.B. First-order phase transitions in
large entropy lattice models. Comm. Math. Phys. 83
(1982), 493–515.
(https://doi.org/10.1007/BF01247839doi:10.1007/BF01247839, 1373377)

Kramers-Wannier
Kramers, H.A. and Wannier, G.H. Statistics of the two-dimensional
ferromagnet. Part I. Phys. Rev. 60 (1941),

252–262.
(https://doi.org/10.1103/PhysRev.60.252doi:10.1103/PhysRev.60.252, 0004803)

KR
Külske,  C. and Rozikov, U.A. Fuzzy transformations and
extremality of Gibbs measures for the Potts model on a Cayley tree.
Random Structures Algorithms 50 (2017),

636–678. (https://doi.org/10.1002/rsa.20671doi:10.1002/rsa.20671,
3660523)


KRK
Külske, C., Rozikov, U.A. and Khakimov, R.M. Description of the
translation-invariant splitting Gibbs measures for the Potts model
on a Cayley tree. J. Stat. Phys. 156 (2014), 189–200.
(https://doi.org/10.1007/s10955-014-0986-ydoi:10.1007/s10955-014-0986-y, 3215122)








Laanait
Laanait, L., Messager, A. and Ruiz, J. Phases coexistence and
surface tensions for the Potts model. Comm. Math. Phys.105 (1986),

527–545. (https://doi.org/10.1007/BF01238932doi:10.1007/BF01238932, 0852089)









Magnus
Magnus, W., Karrass, A. and Solitar, D. Combinatorial Group
Theory: Presentations of Groups in Terms of Generators and
Relations, 2nd revised ed. Dover, New York, 1976. (0422434)



Martin
Martin, P. Potts Models and Related Problems in Statistical
Mechanics. Series on Advances in Statistical Mechanics, 5.
World Scientific, Singapore,

1991. (https://doi.org/10.1142/0983doi:10.1142/0983, 1103994)

Martinelli
Martinelli, F., Sinclair, A. and Weitz, D. Glauber dynamics on
trees: Boundary conditions and mixing time. Comm. Math.Phys. 250 (2004),

301–334.
(https://doi.org/10.1007/s00220-004-1147-ydoi:10.1007/s00220-004-1147-y, 2094519)

Martirosian
Martirosian, D.H. Translation invariant Gibbs states in the
q-state Potts model. Comm. Math. Phys. 105 (1986),

281–290.
(https://doi.org/10.1007/BF01211103doi:10.1007/BF01211103,
0849209)










MezardMézard, M. and Montanari, A. Reconstruction on trees and
spin glass transition. J. Stat. Phys. 124 (2006),
1317–1350.
(https://doi.org/10.1007/s10955-006-9162-3doi:10.1007/s10955-006-9162-3,
2266446)






Mezard-Parisi
Mézard, M. and Parisi, G. The Bethe lattice spin glass revisited.
Eur. Phys. J. B 20 (2001), 217–233.
(https://doi.org/10.1007/PL00011099doi:10.1007/PL00011099, 1832936)

Mos Mossel, E.
Survey: Information flow on trees. In: Graphs, Morphisms and
Statistical Physics (J. Nešetřil and P. Winkler, eds.),
DIMACS Series in Discrete Mathematics and Theoretical Computer
Science 63, 155–170. Amer. Math. Soc., Providence, RI,
2004.
(https://doi.org/10.1090/dimacs/063/12doi:10.1090/dimacs/063/12,
2056226)

Murua
Murua, A. and Quintana, F.A. Semiparametric Bayesian regression via
Potts model. J. Comput. Graph. Statist. 26 (2017),

265–274.
(https://doi.org/10.1080/10618600.2016.1172015doi:10.1080/10618600.2016.1172015,
3640184)

Ost
Ostilli, M. Cayley trees and Bethe lattices: A concise analysis for
mathematicians and physicists. Phys. A 391 (2012),

3417–3423.
(https://doi.org/10.1016/j.physa.2012.01.038doi:10.1016/j.physa.2012.01.038,
2904466)










Peruggi1 Peruggi, F., di Liberto, F. and Monroy, G.
The Potts model on Bethe lattices. I. General results.
J. Phys. A 16 (1983),

811–827.
(https://doi.org/10.1088/0305-4470/16/4/018doi:10.1088/0305-4470/16/4/018,
0706197)


Peruggi2 Peruggi, F., di Liberto, F. and Monroy, G.
Phase diagrams of the q-state Potts model on Bethe lattices.
Phys. A 141 (1987),

151–186.
(https://doi.org/10.1016/0378-4371(87)90267-6doi:10.1016/0378-4371(87)90267-6,
0879715)


Potts Potts, R.B. Some generalized order-disorder
transformations. Proc. Cambridge Philos. Soc. 48
(1952), 106–109.
(https://doi.org/10.1017/S0305004100027419doi:10.1017/S0305004100027419,
0047571)








Pr
Preston, C.J. Gibbs States on Countable Sets. Cambridge
Tracts in Mathematics, 68. Cambridge University Press, London,
1974.
(https://doi.org/10.1017/CBO9780511897122doi:10.1017/CBO9780511897122,
0474556)











Reichardt
Reichardt, J. and Bornholdt, S. Detecting fuzzy community
structures in complex networks with a Potts model. Phys.Rev. Lett. 93 (2004), Art. 218701, 4 pp.
(https://doi.org/10.1103/PhysRevLett.93.218701doi:10.1103/PhysRevLett.93.218701)

Riva
Riva, V. and Cardy, J. Holomorphic parafermions in the Potts model
and stochastic Loewner evolution. J. Stat. Mech. Theory
Exp. (2006), no. 12, P12001, 19 pp.
(https://doi.org/10.1088/1742-5468/2006/12/P12001doi:10.1088/1742-5468/2006/12/P12001, 2280251)


















Ro
Rozikov, U.A. Gibbs Measures on Cayley Trees. World
Scientific, Hackensack, NJ, 2013.
(https://doi.org/10.1142/8841doi:10.1142/8841, 3185400)














Roz-DNA
Rozikov, U.A. Tree-hierarchy of DNA and distribution of Holliday
junctions. J. Math. Biol. 75 (2017), 1715–1733.
(https://doi.org/10.1007/s00285-017-1136-3doi:10.1007/s00285-017-1136-3, 3712328)

RS Rozikov, U.A. and Shoyusupov, Sh.A.  Gibbs
measures for the SOS model with four states on a Cayley tree.
(Russian) Teoret. Mat. Fiz. 149 (2006), no. 1,
18–31; translation in Theor. Math. Phys. 149
(2006), no. 1, 1312–1323.
(https://doi.org/10.1007/s11232-006-0120-7doi:10.1007/s11232-006-0120-7, 2297108)


RoSu Rozikov, U.A. and Suhov, Y.M. Gibbs measures for SOS models on a Cayley
tree. Infin. Dimens. Anal. Quantum Probab. Relat. Top.
9 (2006), 471–488.
(https://doi.org/10.1142/S0219025706002494doi:10.1142/S0219025706002494, 2256506)


Rudin
Rudin, W. Principles of Mathematical Analysis, 3rd ed.
International Series in Pure and Applied Mathematics. McGraw-Hill,
New York, 1976. (0385023)

Sa
Sanyal, S. and Glazier, J.A. Viscous instabilities in flowing
foams: A Cellular Potts Model approach. J. Stat. Mech.Theory Exp. (2006), no. 10, P10008, 9 pp.
(https://doi.org/10.1088/1742-5468/2006/10/P10008doi:10.1088/1742-5468/2006/10/P10008)

Sc Schelling, T.C. Dynamic models of segregation. J. Math. Sociology 1 (1971),
143–186.
(https://doi.org/10.1080/0022250X.1971.9989794doi:10.1080/0022250X.1971.9989794)

Schramm
Schramm, O. Conformally invariant scaling limits: an overview and a
collection of problems. In: Proceedings of the International
Congress of Mathematicians (Madrid, August 22–30, 2006), Vol. I,
513–543. Eur. Math. Soc., Zürich, 2007.
(https://doi.org/10.4171/022-1/20doi:10.4171/022-1/20,
2334202)


Schulze Schulze, C. Potts-like model for ghetto formation in multi-cultural
societies. Int. J. Modern Phys. C 16

(2005), 351–355.
(https://doi.org/10.1142/S0129183105007169doi:10.1142/S0129183105007169)

Shiryaev
Shiryaev, A.N. Probability, 2nd ed. Graduate Texts in
Mathematics, 95. Springer, New York, 1996.
(https://doi.org/10.1007/978-1-4757-2539-1doi:10.1007/978-1-4757-2539-1, 1368405)







Sp
Spitzer, F. Markov random fields on an infinite tree. Ann.Probab. 3 (1975),

387–398.
(https://doi.org/10.1214/aop/1176996347doi:10.1214/aop/1176996347,
0378152)



Sun Sun, L., Chang, Y.F. and Cai, X. A discrete simulation of tumor growth
concerning nutrient concentration. Int. J. Modern Phys. B

18 (2004), 2651–2657.

(https://doi.org/10.1142/S0217979204025853doi:10.1142/S0217979204025853)

Sz
Szabó, A. and Merks, R.M.H. Cellular Potts modeling of tumor
growth, tumor invasion, and tumor evolution. Front. Oncol.3 (2013), Art. 87, 12 pp.
(https://doi.org/10.3389/fonc.2013.00087doi:10.3389/fonc.2013.00087)

Takaishi Takaishi, T. Simulations of financial
markets in a Potts-like model. Int. J. Modern Phys. C 16 (2005),

1311–1317.
(https://doi.org/10.1142/S0129183105007923doi:10.1142/S0129183105007923)

Thouless
Thouless, D.J. Spin-glass on a Bethe lattice. Phys. Rev.Lett. 56 (1986),

1082–1085.
(https://doi.org/10.1103/PhysRevLett.56.1082doi:10.1103/PhysRevLett.56.1082)

Ti
Tikare, V. and Cawley, J.D. Application of the Potts model to
simulation of Ostwald ripening. J. Amer. Ceramic Soc. 81

(1998), 485–491.
(https://doi.org/10.1111/j.1151-2916.1998.tb02366.xdoi:10.1111/j.1151-2916.1998.tb02366.x)

Weiss
Weiss, P.R. The application of the Bethe–Peierls method to
ferromagnetism. Phys. Rev. 74 (1948),

1493–1504.
(https://doi.org/10.1103/PhysRev.74.1493doi:10.1103/PhysRev.74.1493)

Wu Wu, F.Y. The Potts model. Rev. Modern Phys. 54 (1982),  235–268.
(https://doi.org/10.1103/RevModPhys.54.235doi:10.1103/RevModPhys.54.235,
0641370)


Z0 Zachary, S. Countable state space Markov random fields
and Markov chains on trees. Ann. Probab. 11 (1983),

894–903.
(https://doi.org/10.1214/aop/1176993439doi:10.1214/aop/1176993439,
0714953)

















