# Gibbs measures on spatial systems on vertices of Cayley trees

F. H. Haydarov  F.H.Haydarov  New Uzbekistan University, 54, Mustaqillik Ave.,
Tashkent, 100007, Uzbekistan,  Central Asian University, 264, Milliy Bog
street, Tashkent 111221, Uzbekistan,  Institute of mathematics, 9, University
str. Tashkent, 100174, Uzbekistan,  National University of Uzbekistan,
University str 4 Olmazor district, Tashkent, 100174, Uzbekistan.  haydarov_
<EMAIL_ADDRESS>
###### Abstract.

There are many research works devoted to Gibbs measure for models on Cayley
trees. Among these works, there are some works in which the general results
are identical, but the considered models are various. In this article, we
present the construction of Gibbs measures in the language of measure theory
and reply to the question “When can we construct Gibbs specifications?” Also,
we present a new condition (convenient for verification) which is equivalent
to the consistency condition of kernels. The obtained results in the article
are general, not for a particular model. On the contrary, these results hold
for some considered models on Cayley trees.

Mathematics Subject Classifications (2010). 60K35 (primary); 82B05, 82B20
(secondary)

Key words. Cayley tree, spatial systems, kernels, a version of the conditional
expectation, Gibbsian specifications, Ising model.

##  1\. Introduction

The Gibbs measure is a probability measure that describes the equilibrium
state of a physical system on the Cayley tree. In statistical mechanics, the
Gibbs measure is used to describe the equilibrium properties of a system with
many interacting particles. It assigns probabilities to different
configurations of the system based on their energy. The most probable
configurations are those with low energy, and the Gibbs measure gives a way to
calculate these probabilities (e.g. [6, 9]).

Spin models on a graph or in a continuous spaces form a large class of systems
considered in statistical mechanics. Some of them have a real physical
meaning, others have been proposed as suitably simplified models of more
complicated systems. The geometric structure of the graph or a physical space
plays an important role in such investigations. For example, in order to study
the phase transition problem on a cubic lattice $Z^{d}$ or in space
$\mathbb{d}$ one uses, essentially, the Pirogov-Sinai theory; see [15], [16]
[20]. A general methodology of phase transitions in $\mathbb{Z}^{d}$ or
$\mathbb{R}^{d}$ was developed in [11]; some recent results in this direction
have been established in [13] (see also the bibliography therein).

On a Cayley tree, the interactions between particles are often described by
spin models, where each vertex of the tree is associated with a spin variable
that can take on a certain number of values. The spins interact with their
neighbors according to some interaction rule, which determines the energy of a
given configuration (see [17, 18]).

There are many works devoted to several models defined on Cayley trees. But
most of these are written by physicists with little regard for mathematical
rigor. Minority of these papers which contain mathematically rigorous results
about Gibbs measures on Cayley trees. This paper is mainly based on these
mathematical papers. In this paper, we present the construction of Gibbsian
specifications in the language of measure theory and reply to the question
“When can we construct Gibbs specifications?” Also, we present a new condition
(convenient for verification) which is equivalent to the consistency condition
of kernels. The obtained results in the article are general, not for a
particular model. On the contrary, these results hold for some considered
models on Cayley trees.

##  2\. Spatial systems on Cayley trees

The Cayley tree $\Im^{k}=(V,L)$ of order $k\geq 1$ is an infinite tree, i.e.
graph without cycles, each vertex of which has exactly $k+1$ edges. Here $V$
is the set of vertices of $\Im^{k}$ va $L$ is the set of its edges.

Consider models where the spin takes values in the set
$\Phi\subseteq\mathbb{R}_{\infty}^{+}$, and is assigned to the vertices of the
tree. Let $\Omega_{A}=\Phi^{A}$ be the set of all configurations on $A$ and
$\Omega:=\Phi^{V}$. A partial order $\preceq$ on $\Omega$ defined pointwise by
stipulating that $\sigma_{1}\preceq\sigma_{2}$ if and only if
$\sigma_{1}(x)\leq\sigma_{2}(x)$ for all $x\in V$. Thus $(\Omega,\preceq)$ is
a poset, and whenever we consider $\Omega$ as a poset then it will always be
with respect to this partial order. The poset $\Omega$ is complete. If
$\Omega_{1}$ is a non-empty subset of $\Omega$ then its least upper bound is
the mapping given by
$\sup(\Omega_{1})(x)=\sup\\{\sigma(x):\sigma\in\Omega_{1}\\}$ for all $x\in
V$.

We consider all elements of $V$ are numerated (in any order) by the numbers:
$0,1,2,3,...$. Namely, we can write $V=\\{x_{0},x_{1},x_{2},....\\}$.

We denote by $\mathcal{N}$ the set of all finite subsets of $V$. For each
$A\in V$ let $\pi_{A}:\Omega\rightarrow\Phi^{A}$ be given by
$\pi_{A}\left(\sigma_{x}\right)_{x\in V}=\left(\sigma_{x}\right)_{x\in A}$ and
let
$\mathcal{C}_{A}=\pi_{A}^{-1}\left(\mathcal{P}\left(\Phi^{A}\right)\right)$.
Let $\mathcal{C}=\bigcup_{A\in\mathcal{N}}\mathcal{C}_{A}$ and $\mathcal{F}$
is the smallest sigma field containing $\mathcal{C}$.

For a fixed $x^{0}\in V$ we put

$W_{n}=\left\\{x\in V\mid d\left(x,x^{0}\right)=n\right\\},\quad
V_{n}=\bigcup_{m=0}^{n}W_{m},$

where $d(x,y)$ is the distance between the vertices $x$ and $y$ on the Cayley
tree, i.e. the number of edges of the shortest walk (i.e., path) connecting
vertices $x$ and $y$.

For any fixed configuration $\sigma_{A}\in\Omega_{A},\ A\subset V$ we denote:

$\bar{\sigma}_{A}:=\left\\{\sigma\in\Omega:\;\sigma\big{|}_{A}=\sigma_{A}\right\\}.$

######  Corollary 1.

[10]
$\mathcal{F}=\mathcal{S}\left(\left\\{\bar{\sigma}_{V_{n}}:n\in\mathbb{N}_{0}\right\\}\right)$.

A measurable space $(\Omega,\mathcal{F})$, a non-empty index set $V$ equipped
with a partial order $\subseteq$, and a decreasing family
$\mathbb{F}=\left\\{\mathcal{F}^{\Lambda}\right\\}_{\Lambda\in\mathcal{N}}$,
(where $\mathcal{F}^{\Lambda}:=\mathcal{S}(\mathcal{C}_{\Lambda})$) of sub
$\sigma$-fields of $\mathcal{F}$. It is easy to check that the poset
$(\mathcal{N},\subseteq)$ is directed (i.e., for all $A_{1},A_{2}\in D$ there
exists $A\in D$ with $A_{1}\preceq A$ and $A_{2}\preceq A$ ) and countably
generated (i.e., there exists a countable subset $D_{0}$ of $D$ such that for
each $A\in D$ there is an element $A_{0}\in D_{0}$ with $A\preceq A_{0}$ ).

A collection
$\Sigma:=\left(V,\mathcal{N},\Omega,\left\\{\mathcal{F}^{\Lambda}\right\\}_{\Lambda\in\mathcal{N}}\right)$
with $(V,\mathcal{N}),$ $\Omega$ and
$\left\\{\mathcal{F}^{\Lambda}\right\\}_{\Lambda\in\mathcal{N}}$ as above and
with the inclusion order $\subseteq$ on $\mathcal{N}$ countably generated will
be called a _spatial system_. For instance, the lattice and particle models
are both spatial systems. Note that we choose the set of spin values $\Phi$
which is $\mathcal{N}$ is countably generated (see in [10]).

For each $\Lambda\in\mathcal{N}$ let $\mu_{\Lambda}$ be a probability measure
on $\mathcal{F}^{\Lambda}$. Then the family of probability measures
$\left\\{\mu_{\Lambda}\right\\}_{\Lambda\in\mathcal{N}}$ is said to be
_consistent (compatible)_ if $\mu_{\Lambda}(F)=\mu_{\Delta}(F)$ for all
$F\in\mathcal{F}^{\Lambda}$ whenever $\Lambda\subset\Delta$. We say that the
family of $\sigma$-algebras
$\left\\{\mathcal{F}^{\Lambda}\right\\}_{\Lambda\in\mathcal{N}}$ has the
Kolmogorov extension property if for each consistent family
$\left\\{\mu_{\Lambda}\right\\}_{\Lambda\in\mathcal{N}}$ there exists a
probability measure $\mu$ on $\mathcal{F}$ such that $\mu(F)=\mu_{\Lambda}(F)$
for all $F\in\mathcal{F}^{\Lambda},\ \Lambda\in\mathcal{N}$. If $\mu$ exists
then by Carathéodory’s extension theorem it is unique, since it is determined
by the family $\left\\{\mu_{\Lambda}\right\\}_{\Lambda\in\mathcal{N}}$ on the
algebra $\mathcal{C}$ and $\sigma(\mathcal{C})=\mathcal{F}$. Finally, we say
that the spatial system $\Sigma$ has the Kolmogorov extension property if the
family of $\sigma$-algebras
$\left\\{\mathcal{F}^{\Lambda}\right\\}_{\Lambda\in\mathcal{N}}$ does. Let
$\mathcal{N}_{1}$ be a subset of $\mathcal{N}$ which is also directed and
countably generated (cofinal in [9]).

The following theorem gives us any consistent family of probability measures
$\left\\{\mu_{\Lambda}\right\\}_{\Lambda\in\mathcal{N}_{1}}$ extends to a
probability measure $\mu\in\mathrm{P}(\mathcal{F})$.

######  Theorem 1.

Let $\mathcal{N}_{1}$ be a directed and for each $\Lambda\in\mathcal{N}_{1}$
let $\mu_{\Lambda}\in\mathrm{P}\left(\mathcal{F}^{\Lambda}\right)$. If
$\left\\{\mu_{\Lambda}\right\\}_{\Lambda\in\mathcal{N}_{1}}$ is consistent
then there exists a unique probability measure $\mu\in\mathrm{P}(\mathcal{F})$
such that $\mu(F)=\mu_{\Lambda}(F)$ for all
$F\in\mathcal{F}^{\Lambda},\Lambda\in\mathcal{N}_{1}$.

###### Proof.

Let $\mathcal{N}_{1}=\\{\Lambda_{1},\Lambda_{2},...,\Lambda_{s},...\\}$, where
$|\Lambda_{i}|<\infty$ for all $i\in\mathbb{N}$. Since $\mathcal{N}_{1}$ is
directed, there exists $\Lambda^{\prime}_{i}\in\mathcal{N}_{1}$ with
$\Lambda^{\prime}_{i-1}\subseteq\Lambda^{\prime}_{i}$ and
$\Lambda_{i}\subseteq\Lambda^{\prime}_{i}$ where
$\Lambda^{\prime}_{1}=\Lambda_{1}$. Thus, this gives us
$\Lambda^{\prime}_{i}\subseteq\Lambda^{\prime}_{i+1}$ for all
$i\in\mathbb{N}$. Let
$\mathcal{N}^{\prime}_{1}:=\\{\Lambda^{\prime}_{1},\Lambda^{\prime}_{2},...,\Lambda^{\prime}_{s},...\\}$
then for each $\Lambda\in\mathcal{N}$ there is an element
$\Lambda^{\prime}_{q}\in\mathcal{N}^{\prime}_{1}$ with
$\Lambda\subseteq\Lambda^{\prime}_{q}$. Hence, by Kolmogorov extension theorem
there exists a unique probability measure
$\mu^{\prime}\in\mathrm{P}(\mathcal{F}^{\prime})$ such that
$\mu^{\prime}(F)=\mu^{\prime}_{\Lambda^{\prime}_{i}}(F)$ for all
$F\in\mathcal{F}^{\Lambda^{\prime}_{i}},\Lambda^{\prime}_{i}\in\mathcal{N}^{\prime}_{1}$,
where $\mathcal{F}^{\prime}$ is a $\sigma$-field generated by
$\bigcup_{i=1}^{\infty}\mathcal{F}^{\Lambda^{\prime}_{i}}$. Since
$\left\\{\mathcal{F}^{\Lambda}\right\\}_{\Lambda\in\mathcal{N}_{1}}$ is a
decreasing family we get $\mathcal{F}=\mathcal{F}^{\prime}$. By Carathéodory’s
extension theorem we get $\mu^{\prime}=\mu$. ∎

##  3\. Gibbsian specifications on Cayley trees

Note that
$\Sigma=\left(V,\mathcal{N},\Omega,\left\\{\mathcal{F}^{\Lambda}\right\\}_{\Lambda\in\mathcal{N}}\right)$
is a spatial system which defined in previous section. $\mathcal{N}$ equipped
with a directed, countably generated partial order $\subseteq$, and a
decreasing family
$\mathbb{F}=\left\\{\mathcal{F}_{\Lambda}\right\\}_{\Lambda\in\mathcal{N}}$ of
sub-$\sigma$-algebras of $\mathcal{F}$.

Suppose for each $\Lambda\in\mathcal{N}$ we have a strict
$\mathcal{F}_{\Lambda}$-measurable quasi-probability kernel
$\zeta_{\Lambda}\in K(\mathcal{F}_{\Lambda})$. Then the family
$\mathcal{V}=\left\\{\zeta_{\Lambda}\right\\}_{\Lambda\in\mathcal{N}}$ will be
called an $\mathbb{F}$-_specification_ if
$\zeta_{\Delta}=\zeta_{\Delta}\zeta_{\Lambda}$ whenever
$\Lambda,\Delta\in\mathcal{N}$ with $\Lambda\subseteq\Delta$. Let
$\mathcal{V}=\left\\{\zeta_{\Lambda}\right\\}_{\Lambda\in\mathcal{N}}$ be an
$\mathbb{F}$-specification; then a probability measure
$\mu\in\mathrm{P}(\mathcal{F})$ is called a _Gibbs state with specification
$\mathcal{V}$_ if $\mu=\mu\zeta_{\Lambda}$ for each $\Lambda\in\mathcal{N}$.
Note that this definition of Gibbs states originates from Dobrushin [2, 3, 4,
5], and Lanford and Ruelle [12, 19].

######  Definition 1.

Let
$P_{\Lambda}:\Omega\rightarrow\overline{\mathbb{R}}:=\mathbb{R}\cup\\{-\infty,\infty\\}$
be $\mathcal{F}_{\Lambda}$-measurable mapping for all $\Lambda\in\mathcal{N}$,
then the collection $P=\left\\{P_{\Lambda}\right\\}_{\Lambda\in\mathcal{N}}$
is called a potential. Also, the following expression

$H_{\Delta,P}(\sigma)\stackrel{{\scriptstyle\text{ def
}}}{{=}}\sum_{\Delta\cap\Lambda\neq\varnothing,\Lambda\in\mathcal{N}}P_{\Lambda}(\sigma),\quad\forall\sigma\in\Omega.$
(3.1)

is called Hamiltonian $H$ associated to the potential $P$.

Put

$r(P)\stackrel{{\scriptstyle\text{ def
}}}{{=}}\inf\left\\{R>0:P_{\Lambda}\equiv 0\text{ for all }\Lambda\text{ with
}\operatorname{diam}(\Lambda)>R\right\\}.$

If $r(P)<\infty,P$ has finite range and $H_{\Delta;P}$ is well defined. If
$r(P)=\infty$, $P$ has infinite range and, for the Hamiltonian to be well
defined, we will assume that $P$ is absolutely summable in the sense that

$\sum_{\begin{subarray}{c}\Lambda\in\mathcal{N},x\in\Lambda\end{subarray}}\left\|P_{\Lambda}\right\|_{\infty}<\infty,\quad\forall
x\in V,$

(remember that $\left.\|f\|_{\infty}\stackrel{{\scriptstyle\text{ def
}}}{{=}}\sup_{\omega}|f(\omega)|\right)$ which ensures that the interaction of
a spin with the rest of the system is always bounded, and therefore that
$\left\|H_{\Delta;P}\right\|_{\infty}<\infty$.

Let $\mathcal{N}_{1}=\\{V_{n}:n\in\mathbb{N}\\}$ then we define the following
Hamiltonian in the box $V_{n},n\in\mathbb{N}$:

$H_{n}(\sigma)=\sum_{n}P_{V_{n}}(\sigma),\quad\forall\sigma\in\Omega.$ (3.2)

Let us define a specification
$\zeta^{H}=\left\\{\zeta_{V_{n}}^{H}\right\\}_{n\in\mathbb{N}}$ (in short
$\zeta_{V_{n}}^{H}:=\zeta_{n}^{H}$) such that
$\zeta_{V_{n}}^{H}(\cdot\mid\omega)$ gives to each configuration
$\tau_{V_{n}}\omega_{V_{n}}^{c}$ a probability proportional to the Boltzmann
weight prescribed by equilibrium statistical mechanics (see e.g. [6]):

$\zeta_{n}^{P}\left(\omega,\sigma_{n}\right)\stackrel{{\scriptstyle\text{ def
}}}{{=}}\frac{1}{\mathbf{Z}_{n}^{\omega}}e^{-H_{n}\left(\sigma_{n}\omega_{\bar{V}_{n}}\right)},$
(3.3)

where we have written explicitly the dependence on $\omega_{\bar{V}_{n}}$, and
$\mathbf{Z}_{n}^{\omega}$ is a partition function, i.e.,

$\mathbf{Z}_{n}^{\omega}\stackrel{{\scriptstyle\text{ def
}}}{{=}}\sum_{\sigma_{n}\in\Omega_{V_{n}}}\exp\left(-H_{n}\left(\sigma_{n}\omega_{\bar{V}_{n}}\right)\right).$

######  Theorem 2.

If $\mathbb{F}_{1}:=\left\\{\zeta_{n}^{P}\right\\}_{n\in\mathbb{N}}$ then
$\mathbb{F}_{1}$ is a specification.

###### Proof.

For a fixed $m,n\in\mathbb{N}$ with $V_{n}\subset V_{m}\Subset V$, we show
that $\zeta_{m}^{P}\zeta_{n}^{P}=\zeta_{m}^{P}$. Let
$A\subset\mathcal{P}(\Omega_{V_{n}})$. At first, we consider the case
$A=\\{\sigma_{V_{n}}\\}$, i.e.,

$\displaystyle\zeta_{m}\zeta_{n}(\omega,\sigma_{V_{n}})$
$\displaystyle=\sum_{\tau_{V_{m}}}\zeta_{m}\left(\omega,\tau_{V_{m}}\right)\zeta_{n}\left(\tau_{V_{m}}\omega_{V_{m}^{c}},\sigma_{V_{n}}\right)=\sum_{\tau_{V_{m}}}\zeta_{m}\left(\omega,\tau_{V_{m}}\right)\zeta_{n}\left(\tau_{V_{m}\backslash
V_{n}}\omega_{V_{m}^{c}},\sigma_{V_{n}}\right).$

From the last equation, for any $A\subset\mathcal{P}(\Omega_{V_{n}})$ we
obtain

$\displaystyle\zeta_{m}\zeta_{n}(A\mid\omega)=\sum_{\tau_{V_{m}}}\sum_{\eta_{\Delta}}\mathbf{1}_{A}\left(\eta_{V_{n}}\tau_{V_{m}\backslash
V_{n}}\omega_{V_{m}^{c}}\right)\zeta_{m}\left(\omega,\tau_{V_{m}}\right)\zeta_{n}\left(\tau_{V_{m}\backslash
V_{n}}\omega_{V_{m}^{c}},\eta_{V_{n}}\right).$

For any configuration $\tau_{V_{m}}$ which defined on $V_{m}$ we rewrite this
configuration as a combination of $\tau_{V_{n}}^{\prime}$ on $V_{n}$ and
$\tau_{V_{m}\backslash V_{n}}$ on $V_{m}\backslash V_{n}$, i.e.
$\tau_{V_{m}}=\tau_{V_{n}}^{\prime}\vee\tau_{V_{m}\backslash
V_{n}}^{\prime\prime}$. By (3.3), RHS of the last equation can be rewritten
as:

$\sum_{\tau_{V_{m}\backslash
V_{n}}^{\prime\prime}}\sum_{\eta_{V_{n}}}\mathbf{1}_{A}\left(\eta_{V_{n}}\tau^{\prime\prime}_{V_{m}\backslash
V_{n}}\omega_{V_{m}^{c}}\right)\frac{e^{-H_{V_{n}}\left(\eta_{V_{n}}\tau^{\prime\prime}_{V_{m}\backslash
V_{n}}\omega_{V_{m}^{c}}\right)}}{\mathbf{Z}_{m}\left(\omega_{V_{m}^{c}}\right)\mathbf{Z}_{n}\left(\tau_{V_{m}\backslash
V_{n}}^{\prime\prime}\omega_{V_{m}^{c}}\right)}\sum_{\tau_{V_{n}}^{\prime}}e^{-H_{V_{m}}\left(\tau_{V_{n}}^{\prime}\tau_{V_{m}\backslash
V_{n}}^{\prime\prime}\omega_{V_{m}^{c}}\right)}.$

It’s easy to check that the following expression

$H_{m}\left(\tau_{V_{n}}^{\prime}\tau_{V_{m}\backslash
V_{n}}^{\prime\prime}\omega_{V_{m}^{c}}\right)-H_{n}\left(\tau_{V_{n}}^{\prime}\tau_{V_{m}\backslash
V_{n}}^{\prime\prime}\omega_{V_{m}^{\mathrm{c}}}\right)$

does not depend on $\tau_{V_{n}}^{\prime}$. That’s why we have

$H_{m}\left(\tau_{V_{n}}^{\prime}\tau_{V_{m}\backslash
V_{n}}^{\prime\prime}\omega_{V_{m}^{c}}\right)-H_{n}\left(\tau_{V_{n}}^{\prime}\tau_{V_{m}\backslash
V_{n}}^{\prime\prime}\omega_{V_{m}^{\mathrm{c}}}\right)=H_{m}\left(\eta_{V_{n}}\tau_{V_{m}\backslash
V_{n}^{\prime\prime}}\omega_{V_{m}^{\mathrm{c}}}\right)-H_{n}\left(\eta_{V_{n}}\tau_{V_{m}\backslash
V_{n}^{\prime\prime}}\omega_{V_{m}^{\mathrm{c}}}\right),$

which gives

$\sum_{\tau_{V_{n}}^{\prime}}e^{-H_{m}\left(\tau_{V_{n}}^{\prime}\tau_{V_{m}\backslash
V_{n}}^{\prime\prime}\omega_{V_{m}^{c}}\right)}=\mathbf{Z}_{n}\left(\tau^{\prime\prime}_{V_{m}\backslash
V_{n}}\omega_{V_{m}^{c}}\right)e^{H_{m}\left(\eta_{V_{n}}\tau_{V_{m}\backslash
V_{n}}^{\prime\prime}{}^{\omega}{}^{c}\right)}e^{-H_{n}\left(\eta_{V_{n}}\tau_{V_{m}\backslash
V_{n}}^{\prime\prime}\omega_{V_{m}^{c}}\right)}.$

Inserting this in the above expression, and renaming
$\eta_{V_{n}}\tau_{V_{m}\backslash
V_{n}}^{\prime\prime}\equiv\eta_{V_{m}}^{\prime}$, we get

$\zeta_{m}\zeta_{n}(A\mid\omega)=\sum_{\eta_{V_{m}}^{\prime}}\mathbf{1}_{A}\left(\eta_{V_{m}}^{\prime}\omega_{V_{m}^{c}}\right)\frac{e^{-H_{m}\left(\eta_{V_{m}}^{\prime}\omega_{V_{m}c}\right)}}{\mathbf{Z}_{m}\left(\omega_{V_{m}^{c}}\right)}=\zeta_{m}(A\mid\omega).$

∎

Now we give more comfortable condition for special kernels which is equivalent
to the condition of consistency.

We consider some models discussed in this section with the corresponding
potentials on Cayley trees:

$\bar{P}_{B}(\sigma)=\begin{cases}-\beta\rho(\sigma(x),\sigma(y))&\text{ if
}B=\langle x,y\rangle,\\\ 0&\text{ otherwise.}\end{cases}$ (3.4)

Let
$h:(\Omega,\mathcal{P}(\Omega))\rightarrow(\mathbb{R},\mathcal{B}(\mathbb{R}))$
be a measurable mapping. We define a kernel:

$\zeta_{n}^{\bar{P}}(h,\sigma_{n})=\frac{1}{\mathbf{Z}^{h}_{n}}\exp\left\\{J\beta\sum_{V_{n}\cap\Lambda\neq\varnothing,\Lambda\in\mathcal{N}}\bar{P}_{\Lambda}(\sigma_{n})+\sum_{x\in
W_{n}}\rho(\sigma_{n}(x),h(x))\right\\},$ (3.5)

where

$\mathbf{Z}^{h}_{n}=\sum_{\sigma_{n}\in\Omega_{V_{n}}}\exp\left\\{J\beta\sum_{V_{n}\cap\Lambda\neq\varnothing,\Lambda\in\mathcal{N}}\bar{P}_{\Lambda}(\sigma_{n})+\sum_{x\in
W_{n}}\rho(\sigma_{n}(x),h(x))\right\\}.$

It’s easy to check that $\zeta_{n}^{\bar{P}}(\sigma_{n},h)$ satisfies the
conditions of kernel.

Are the family of kernels
$\\{\zeta_{n}^{\bar{P}}(\sigma_{n},h)\\}_{n\in\mathbb{N}}$ specification or
not? For a fixed $m,n\in\mathbb{N}$ with $V_{n}\subset V_{m}\Subset V$, we
show that $\zeta_{m}^{\bar{P}}\zeta_{n}^{\bar{P}}=\zeta_{m}^{\bar{P}}$. But it
is easy to check the condition. From Corollary 1 it is enough to check the
condition $\zeta_{n+1}^{\bar{P}}\zeta_{n}^{\bar{P}}=\zeta_{n+1}^{\bar{P}}$.
Also, the last equality is equivalent to the following equality:

$\sum_{\omega_{n}\in\\{-1,1\\}^{W_{n}}}\zeta_{n}^{\bar{P}}(\sigma_{n-1}\omega_{n},h)=\zeta_{n-1}^{\bar{P}}(\sigma_{n-1},h).$
(3.6)

We show that more comfortable condition which is equivalent to (3.6).

######  Theorem 3.

Probability distributions $\zeta_{n}^{\bar{P}}(h,\sigma_{n}),n=1,2,\ldots$, in
(3.5) are compatible if and only if for any $x\in V$ the following equation
holds:

$\prod_{y\in
S(x)}\frac{\sum_{u\in\\{-1,1\\}}\exp\left(J\beta\rho(1,u)+\rho(u,h_{y})\right)}{\sum_{u\in\\{-1,1\\}}\exp\left(J\beta\rho(-1,u)+\rho(u,h_{y})\right)}=\exp\\{\rho(1,h_{x})-\rho(-1,h_{x})\\}.$
(3.7)

###### Proof.

Necessity. Suppose that (3.6) holds; we want to prove (3.7). Substituting
(3.5) in (3.6), obtain that for any configurations $\sigma_{n-1}:x\in
V_{n-1}\mapsto\sigma_{n-1}(x)\in\\{-1,1\\}$ :

From (3.4) we can rewrite (3.5) as

$\zeta_{n}^{P}(h,\sigma_{n})=\frac{1}{\mathbf{Z}^{h}_{n}}\exp\left\\{J\beta\sum_{\langle
x,y\rangle,x,y\in V_{n}}\rho(\sigma_{n}(x),\sigma_{n}(y))+\sum_{x\in
W_{n}}\rho(\sigma_{n}(x),h(x))\right\\},$ (3.8)
$\displaystyle\frac{\mathbf{Z}^{h}_{n-1}}{\mathbf{Z}^{h}_{n}}\sum_{\omega_{n}\in\Omega_{W_{n}}}\exp\left(\sum_{x\in
W_{n-1}}\sum_{y\in
S(x)}\left(J\beta\rho(\sigma_{n-1}(x),\omega_{n}(y))+\rho(\omega_{n}(y),h_{y})\right)\right)=$
(3.9) $\displaystyle\exp\left(\sum_{x\in
W_{n-1}}\rho(\sigma_{n-1}(x),h_{x})\right),$

where $\omega_{n}:x\in W_{n}\mapsto\omega_{n}(x)$. From (3.9) we get:

$\begin{gathered}\frac{\mathbf{Z}^{h}_{n-1}}{\mathbf{Z}^{h}_{n}}\sum_{\omega_{n}\in\Omega_{W_{n}}}\prod_{x\in
W_{n-1}}\prod_{y\in
S(x)}\exp\left(J\beta\rho(\sigma_{n-1}(x),\omega_{n}(y))+\rho(\omega_{n}(y),h_{y})\right)=\\\
\prod_{x\in
W_{n-1}}\exp\left(\rho(\sigma_{n-1}(x),h_{x})\right).\end{gathered}$

Rewrite now the last equality for $\sigma_{n-1}(x)=1$ and
$\sigma_{n-1}(x)=-1$, then dividing first of them to the second one we get

$\prod_{y\in
S(x)}\frac{\sum_{u\in\\{-1,1\\}}\exp\left(J\beta\rho(1,u)+\rho(u,h_{y})\right)}{\sum_{u\in\\{-1,1\\}}\exp\left(J\beta\rho(-1,u)+\rho(u,h_{y})\right)}=\exp\\{\rho(1,h_{x})-\rho(-1,h_{x})\\}.$

Sufficiency. Suppose that (3.7) holds. It is equivalent to the representations

$\prod_{y\in
S(x)}\sum_{u\in\\{-1,1\\}}\exp\left(J\beta\rho(t,u)+\rho(u,h_{y})\right)=a(x)\exp\left(\rho(t,h_{x})\right),t\in\\{-1,1\\}$
(3.10)

for some function $a(x)>0,x\in V$. We have

$\begin{gathered}\text{LHS of}\
(\ref{2.7})=\frac{1}{\mathbf{Z}^{h}_{n}}\exp\left(J\beta\sum_{\langle
x,y\rangle,x,y\in
V_{n-1}}\rho(\sigma_{n-1}(x),\sigma_{n-1}(y))\right)\times\\\ \prod_{x\in
W_{n-1}}\prod_{y\in
S(x)}\sum_{u\in\\{-1,1\\}}\exp\left(J\beta\rho(\sigma_{n-1}(x),u)+\rho(u,h_{y})\right).\end{gathered}$
(3.11)

Substituting (3.10) into (3.11) and denoting $\mathbf{A}^{h}_{n-1}=\prod_{x\in
W_{n-1}}a(x)$, we get

$\begin{gathered}\text{RHS of}\
(\ref{2.12})=\frac{\mathbf{A}^{h}_{n-1}}{\mathbf{Z}^{h}_{n}}\exp\left(J\beta\sum_{\langle
x,y\rangle,x,y\in
V_{n-1}}\rho(\sigma_{n-1}(x),\sigma_{n-1}(y))\right)\times\\\ \prod_{x\in
W_{n-1}}\exp\left(\rho(\sigma_{n-1}(x),h_{x})\right).\end{gathered}$ (3.12)

Since $\zeta_{n}^{\bar{P}},n\geq 1$ is a probability, we should have

$\sum_{\sigma_{n-1}\in\Omega_{V_{n-1}}}\sum_{\omega_{n}\in\Omega_{W_{n}}}\zeta_{n}^{\bar{P}}\left(\sigma_{n-1}\omega_{n},h\right)=1$

Hence from (3.12) we get
$\mathbf{Z}^{h}_{n-1}\mathbf{A}^{h}_{n-1}=\mathbf{Z}^{h}_{n}$, and (3.6)
holds. ∎

Application to Ising model. One of classic models in statistical mechanics is
Ising model. We give above results for Ising model. Given a finite volume
$\Lambda$ in $\Im^{k}$, for a prescribed boundary condition (b.c.)
$\omega_{\Lambda^{c}}\in\Omega_{\Lambda^{c}}=$ $\\{-1,1\\}^{\Lambda^{c}}$, we
define Hamiltonians on $\Omega$ for any $\sigma\in\Omega$ to be the uniformly
convergent series

$H_{\Lambda}^{\omega}\left(\sigma_{\Lambda}\right)=-\sum_{\begin{subarray}{c}x,y\in\Lambda\\\
\langle
x,y\rangle\end{subarray}}J\sigma_{x}\sigma_{y}-\sum_{\begin{subarray}{c}x\in\partial(\Lambda),y\in
V\setminus\Lambda\\\ \langle x,y\rangle\end{subarray}}J\sigma_{x}\omega_{y}.$

For a fixed inverse temperature $\beta>0$, the Gibbs specification is
determined by a family of probability kernels
$\zeta=\left(\zeta_{\Lambda}\right)_{\Lambda\in\mathcal{S}}$ defined on
$\Omega_{\Lambda}\times\mathcal{F}_{\Lambda^{c}}$ by the Boltzmann-Gibbs
weights

$\zeta_{\Lambda}(\sigma\mid\omega)=\frac{1}{Z_{\Lambda}^{\omega}}e^{-\beta
H_{\Lambda}^{\omega}\left(\sigma_{\Lambda}\right)}$ (3.13)

where $Z_{\Lambda}^{\omega}=\sum_{\sigma\in\Omega_{\Lambda}}e^{-\beta
H_{\Lambda}^{\omega}\left(\sigma_{\Lambda}\right)}$ is the partition function,
related to free energy.

Note that by Theorem 2 we can conclude that the family of kernels
$\\{\zeta_{\Lambda}(\sigma\mid\omega)\\}_{\lambda\in\mathcal{N}}$ satisfies
conditions of specification. If we take
$\rho{\sigma,\omega}=\sigma\cdot\omega$ and $h_{x}=J\sum_{y\in S(x)}\omega(y)$
then from Theorem 3 the consistency condition will be equivalent to the
following euqation:

$\prod_{y\in S(x)}\frac{\sum_{u\in\\{-1,1\\}}\exp\left(J\beta
u+h_{y}u\right)}{\sum_{u\in\\{-1,1\\}}\exp\left(-J\beta
u+h_{y}u\right)}=\exp\left(2h_{x}\right).$

If we define new function

$f(h,\theta)=\frac{1}{2}\ln\frac{(1+\theta)e^{2h}+(1-\theta)}{(1-\theta)e^{2h}+(1+\theta)}.$

the the last equation can be rewritten as

$h_{x}=\sum_{y\in S(x)}f\left(h_{y},\theta\right).$

Namely, we give well-known theorem for Ising models on Cayley trees (see e.g.
[18]).

######  Corollary 2.

On Cayley trees the probability kernels
$\\{\zeta_{\Lambda}(\sigma\mid\omega)\\}_{\lambda\in\mathcal{N}}$ for Ising
model are consistent iff for any $x\in V$ the following equation holds:

$h_{x}=\sum_{y\in S(x)}f\left(h_{y},\theta\right).$

Each solution to the equation $h_{x}=\sum_{y\in
S(x)}f\left(h_{y},\theta\right)$ gives us a splitting Gibbs measure. If we
choose $\rho$ as a Kronecker symbol, then the obtained model is called Potts
model. Our results hold for Potts model (see detail in [7, 8, 14, 18]).

## Acknowledgements

The work supported by the fundamental project (number: F-FA-2021-425) of The
Ministry of Innovative Development of the Republic of Uzbekistan.

## Statements and Declarations

Conflict of interest statement: The author states that there is no conflict of
interest.

## Data availability statements

The datasets generated during and/or analysed during the current study are
available from the corresponding author on reasonable request.

## References

  * [1] Athreya K.B., Lahiri S.N.: Measure theory and probability theory. Springer, (2006).
  * [2] Dobrushin, R.L.: Description of a random field by means of conditional probabilities and the conditions governing its regularity. Theory Prob. and its Appl., 13, 197-224, (1968).
  * [3] Dobrushin, R.L.: Gibbsian random fields for lattice systems with pairwise interactions. Funct. Anal. and its Appl., 2, 292-301, (1968).
  * [4] Dobrushin, R.L.: The problem of uniqueness of a Gibbsian random field and the problem of phase transition. Funct. Anal. and its Appl., 2, 302-312, (1968).
  * [5] Dobrushin, R.L. (1969): Gibbsian random fields. The general case. Funct. Anal. and its Appl., 3, 32-28, (1969).
  * [6] Friedli S., Velenik Y.: Statistical Mechanics of Lattice Systems. Cambridge University Press, (2017).
  * [7] Gandolfo D., Haydarov F.H., Rozikov U.A., and Ruiz J.: New Phase Transitions of the Ising Model on Cayley Trees. J. Stat. Phys. 153: 400–411 (2013).
  * [8] Ganikhodjaev N.N.: Limiting Gibbs measures of Potts model with countable set of spin values J. Math. Anal. Appl., 336, 693-703, (2007).
  * [9] Georgii H.O.: Gibbs Measures and Phase Transitions. de Gruyter Studies in Mathematics, 9, (2011).
  * [10] Haydarov F.H.: Kolmogorov extension theorem for non-probability measures on Cayley trees. arXiv:2305.15443v1, (2023).
  * [11] Kotecky, R. and Shlosman, S.B. : First-order phase transition in large entropy lattice models. Commun. Math. Phys. 83 (1982), 493-515.
  * [12] Landford, O.E., Ruelle, D.: Observables at infinity and states with short range correlations in statistical mechanics. Comm. Math. Physics, 13, 194-215, (1968).
  * [13] Mazel, A., Suhov, Y., Stuhl, I. : A classical WR model with $q$ particle types. J.Stat.Phys. 159 (2015), 1040-1086.
  * [14] Mukhamedov F., Barhoumi A., Souissi A., El Gheteb S.: A quantum Markov chain approach to phase transitions for quantum Ising model with competing XY-interactions on a Cayley tree. J. Math. Phys. 61, (2020).
  * [15] Pigorov, S.A., Sinai, Ya.G. : Phase diagrams of classical lattice systems (Russian). Theor. and Math. Phys. 25 (1975), 358-369.
  * [16] Pigorov, S.A., Sinai, Ya.G. : Phase diagrams of classical lattice systems. Continuation (Russian). Theor. and Math. Phys. 26 (1976), 61-76.
  * [17] Rozikov U., Haydarov F.: Invariance Property on Group Representations of the Cayley Tree and Its Applications. Results in Mathematics, 77(6), (2022).
  * [18] Rozikov U.: Gibbs measures on Cayley trees. World Scientific, (2013).
  * [19] Ruelle, D.: Superstable interactions in statistical mechanics. Comm. Math. Physics, 18, 127-159, (1970).
  * [20] Sinai, Ya.G. : Theory of phase transitions: Rigorous Results (Pergamon, Oxford, 1982).