On a $p$-Adic Generalized Gibbs Measure for Ising Model on a Cayley Tree

TL;DR

This paper investigates $p$-adic Ising models on Cayley trees, providing a complete description of translation-invariant Gibbs measures for a specific case and demonstrating phase transitions for certain primes and tree orders.

Contribution

It offers a full characterization of $p$-adic Gibbs measures for the Ising model on Cayley trees and establishes conditions for phase transitions based on prime congruences.

Findings

Complete description of translation-invariant Gibbs measures for $k=3$

Existence of phase transition for $p$-adic Ising models when $p \\equiv 1 \\operatorname{mod }4$ and $k \\geq 3$

Phase transition phenomena depend on prime congruences and tree order

Abstract

In this paper we consider a $p$-adic Ising model on the Cayley tree of order $k\geq 2$. We give full description of all $p$-adic translation-invariant generalized Gibbs measures for $k=3$. Moreover, we show the existence of phase transition for $p$-adic Ising model for any $k\geq3$ when $p\equiv1(\operatorname{mod }4)$.

Equations

$$|x|_{p}=\left\{\begin{array}[]{ll}p^{-r},&\mbox{if}\ \,x\neq 0,\\ 0,&\mbox{if}\ \,x=0.\end{array}\right.$$

$$x=p^{\gamma (x)}(x_0+x_{1}p+x_2{p}^2+...)$$

$$B(a,r)=\{x\in {\mathbb{Q}}_p:|x-a{|}_p\le r\}.$$

$$\text{mbox}log{}_p(x)={\log }_p\left(1+(x-1)\right)=\sum _{n=1}^{\infty }{(-1)}^{n+1}\frac{{(x-1)}^n}{n!},$$

$${\exp }_p(x)=\sum _{n=0}^{\infty }\frac{x^n}{n!},$$

$${\mathcal{E}}_p=\left\{x\in {\mathbb{Q}}_p:|x-1{|}_p<p^{-1/(p-1)}\right\}.$$

$$\mu \left(\bigcup _{j=1}^n{A}_j\right)=\sum _{j=1}^n\mu (A_j).$$

$$W_n=\{x\in V:|x|=n\},V_n=\{x\in V:|x|\le n\},L_n=\{l=<x,y>\in L:x,y\in V_n\}$$

$$S(x)=\{y\in V:x\to y\},S_1(x)=\{y\in V:d(x,y)=1\}.$$

$$({\sigma_{n-1}}\vee{\varphi^{(n)}})(x)=\left\{\begin{array}[]{ll}{\sigma_{n-1}}(x),&\mbox{if}\,\,x\in{V_{n-1}}.\\ {\varphi^{(n)}}(x),&\mbox{if}\,\,x\in{W_{n}}.\end{array}\right.$$

$$H(\sigma )=J\sum _{⟨x,y⟩\in L}\sigma (x)\sigma (y),$$

$${\mu }_h^{(n)}({\sigma }_n)=Z_{n,h}^{-1}{\exp }_p\{H_n({\sigma }_n)\}\prod _{x\in W_n}{h}_x^{({\sigma }_n)},n=1,2,\dots ,$$

$$Z_{n,h}=\sum _{\phi \in {\Omega }_{V_n}}{\exp }_p\{H_n(\phi )\}\prod _{x\in W_n}{h}_x^{\phi (x)},$$

$$H_n({\sigma }_n)=J\sum _{⟨x,y⟩\in L_n}{\sigma }_n(x){\sigma }_n(y).$$

$$\sum _{\phi \in {\Omega }_{W_n}}{\mu }_h^{(n)}({\sigma }_{n-1}\vee \phi )={\mu }_h^{(n-1)}({\sigma }_{n-1}.)$$

$$h_x^2=\prod _{y\in S(x)}\frac{\theta h_y^2+1}{h_y^2+\theta },$$

$$h^2={\left(\frac{\theta h^2+1}{h^2+\theta }\right)}^3.$$

$$z={\left(\frac{\theta z+1}{z+\theta }\right)}^3,$$

$$(z^2-1)\left(z^2+(3\theta -{\theta }^3)z+1\right)=0.$$

$$z_{3,4}=\frac{{\theta }^3-3\theta \pm ({\theta }^2-1)\sqrt{{\theta }^2-4}}{2}.$$

$$z_3+1$$

$$\mathcal{N}_{p}=\left\{\begin{array}[]{ll}2,&\mbox{if }\ p\not\equiv 1(\operatorname{mod}4)\\ 4,&\mbox{if }\ p\equiv 1(\operatorname{mod}4),p\not\equiv 1(\operatorname{mod}3)\\ 8,&\mbox{if }\ p\equiv 1(\operatorname{mod}12)\end{array}\right.$$

$$|TIpGGM|=\left\{\begin{array}[]{ll}1,&\mbox{if }\ p\not\equiv 1(\operatorname{mod}4)\\ 2,&\mbox{if }\ p\equiv 1(\operatorname{mod}4),p\equiv 2(\operatorname{mod}3)\\ 4,&\mbox{if }\ p\equiv 1(\operatorname{mod}12)\end{array}\right.$$

$$Z_{n+1,h}=A_{n,h}{Z}_{n,h},$$

$$A_{n,h}={\left(\frac{\left(\theta h^2+1\right)\left(h^2+\theta \right)}{\theta h^2}\right)}^{2^{n-1}}.$$

$$Z_{n,h_1}=A_{n-1,h_1}{Z}_{n-1,h_1}=\cdots =A_{n-1,h_1}{A}_{n-2,h_1}\dots A_{1,h_1}{Z}_{1,h_1}={\left(\frac{{\left(\theta +1\right)}^2}{\theta }\right)}^{2^{n-1}-1}.$$

$$Z_{n,h_2}={\left(\frac{\left(\theta h_2^2+1\right)\left(h_2^2+\theta \right)}{\theta h_2^2}\right)}^{2^{n-1}-1}={\left(\frac{{\left(\theta -1\right)}^2}{\theta }\right)}^{2^{n-1}-1}.$$

$$|h_3^2+1{|}_p$$

$$Z_{n,h_3}={\left(\frac{\left(\theta h_3^2+1\right)\left(h_3^2+\theta \right)}{\theta h_3^2}\right)}^{2^{n-1}-1}={\left(\frac{\left(\theta (z_3+1)+1-\theta \right)\left(z_3+1+\theta -1\right)}{\theta z_3}\right)}^{2^{n-1}-1}.$$

$${\left|Z_{n+1,h_3}\right|}_p\le {\left|\theta -1\right|}_p^{2^n-2}.$$

Full text

Key words: $p$-adic numbers, Ising model, Gibbs measure, phase transition.

1. Introduction

One of the central problems in statistical mechanics is the study of infinite-volume Gibbs measures corresponding to a given Hamiltonian. This problem includes the study of phase transitions: it occurs for a Hamiltonian if there exist at least two distinct Gibbs measures. However, a complete analysis of the set of all Gibbs measures for a specific Hamiltonian is often a difficult problem. For this reason, most of the work on this topic is devoted to the study of Gibbs measures on Cayley trees [2], [5].

It is known [12], [26] that $p$-adic models in physics can not be described using the usual probability theory. In [12] an abstract $p$-adic probability theory was developed by the theory of non-Archimedean measures. Probabilistic processes in the field of $p$-adic numbers have been studied by many authors (see [1], [17], [24]). In [4] non-Archimedean analogue of Kolmogorov s theorem is proved.

We note that $p$-adic Gibbs measures were studied for several $p$-adic models of statistical mechanics [3, 6, 11, 13, 14, 16, 19, 20, 21, 22, 7, 8, 9, 23]. In [10] it has been proven that for a $\lambda$ -model on a Cayley tree of order $k$ there is no phase transitions. Recall [11] that the Ising model is a special case of the $\lambda$ -model.

In [27], [28] authors constructed new set real values Gibbs measures for the Ising model on the Cayley tree. In [29] for the Ising model on the Cayley tree of order $k$, a new sets of non translation-invariant ($(k_{0})-$translation-invariant) Gibbs measures is constructed.

In this paper we study existence phase transition for the $p$-adic Ising model on the Cayley tree of order three. Using the results of [8] by ART construction we give new $p$-adic generalized Gibbs measures for the Ising model and show the existence of phase transition for arbitrary $k\geq 3$.

2. Preliminaries

2.1. $p$-adic Numbers and Measures

Let $\mathbb{Q}$ be the field of rational numbers. For a fixed prime number $p$, every rational number $x\neq 0$ can be represented in the form $x={p^{r}}\frac{n}{m}$ where, $r,n\in{\mathbb{Z}}$, $m$ is a positive integer, and $n$ and $m$ are relatively prime with $p$. The $p$-adic norm of $x$ is given by

[TABLE]

This norm is non-Archimedean, i.e. it satisfies the strong triangle inequality $|x+y|_{p}\leq\max\{|x|_{p},|y|_{p}\}$ for all $x,y\in\mathbb{Q}$. From this property immediately follow the following facts:

1. if $|x{|_{p}}\neq|y{|_{p}}$, then $|x-y{|_{p}}=\max\{|x{|_{p}},|y{|_{p}}\};$

2. if $|x{|_{p}}=|y{|_{p}}$, then $|x-y{|_{p}}=|x{|_{p}}$.

The completion of $\mathbb{Q}$ with respect to the $p$-adic norm defines the $p$-adic field $\mathbb{Q}_{p}$ (see [15]).

The completion of the field of rational numbers $\mathbb{Q}$ is either the field of real numbers $\mathbb{R}$ or one of the fields of $p$-adic numbers $\mathbb{Q}_{p}$ (Ostrowski’s theorem).

Any $p$-adic number $x\neq 0$ can be uniquely represented in the canonical form

[TABLE]

where $\gamma=\gamma(x)\in{\mathbb{Z}}$ and the integers ${x_{j}}$ satisfy: ${x_{0}}>0,\,\,0\leq{x_{j}}\leq p-1$ ( see [15], [25], [26]). In this case $|x{|_{p}}\,={p^{-\gamma(x)}}$.

Theorem 2.2**.**

[26*]** The equation ${x^{2}}=a,\,\,0\neq a={p^{\gamma(a)}}({a_{0}}+{a_{1}}p+{a_{2}}{p^{2}}+...),\,\,0\leq{a_{j}}\leq p-1,\,{a_{0}}>0$ has a solution in $x\in\mathbb{Q}_{p}$ iff hold true the following:

i) $\gamma(a)$ is even;

ii) ${x^{2}}\equiv{a_{0}}(\mbox{mod}\,p)$ is solvable for $p\neq 2$; the equality ${a_{1}}={a_{2}}=0$ hold if $p=2$.*

Corollary 2.3**.**

[26]** The equation ${x^{2}}=-1$ has a solution in $\mathbb{Q}_{p}$, if and only if $p\equiv 1(\operatorname{mod}4)$.

For $a\in\mathbb{Q}_{p}$ and $r>0$ we denote

[TABLE]

$p$-Adic logarithm is defined by the series

[TABLE]

which converges for $x\in B(1,1)$ and $p$-adic exponential is defined by

[TABLE]

which converges for $x\in B\left({0,{p^{-1/(p-1)}}}\right).$

We set

[TABLE]

This set is the range of the $p$-adic exponential function. It is known [7] the following fact.

Lemma 2.4**.**

*Let $p\geq 3$. Then the set $\mathcal{E}_{p}$ has the following properties:

$(a)$ $\mathcal{E}_{p}$ is a group under multiplication;

$(b)$ $|a-b|_{p}<1$ for all $a,b\in\mathcal{E}_{p}$;

$(c)$ $|a+b|_{p}=1$ for all $a,b\in\mathcal{E}_{p}$;

$(d)$ If $a\in\mathcal{E}_{p}$, then there is an element $h\in B_{p^{-1/(p-1)}}(0)$ such that $a=\exp_{p}(h)$.*

A more detailed description of $p$-adic calculus and $p$-adic mathematical physics can be found in [15], [25], [26].

Let $(X,\mathcal{B})$ be a measurable space, where $\mathcal{B}$ is an algebra of subsets $X$. A function $\mu:\mathcal{B}\to\mathbb{Q}_{p}$ is said to be a $p$-adic measure if for any ${A_{1}},{A_{2}},...,{A_{n}}\in\mathcal{B}$ such that

${A_{i}}\cap{A_{j}}=\emptyset,\,\,i\neq j$, the following holds:

[TABLE]

A $p$-adic measure $\mu$ is called bounded if $\sup\{|\mu(A){|_{p}}:A\in\mathcal{B}\}<\infty$ (see, [12]). It is said that $p$-adic measure is probabilistic if $\mu(X)=1$ [4].

2.5. Cayley tree

The Cayley tree $\Gamma^{k}$ of order $k\geq 1$ is an infinite tree i.e., a graph without cycles, such that exactly $k+1$ edges originate from each vertex. Denote by $V$ the set of vertices, and by $L$ the set of edges of the Cayley tree $\Gamma^{k}$. Two vertices $x$ and $y$ are called nearest neighbours if there exist an edge $l\in L$ connecting them and denote by $l=\langle x,y\rangle.$

Fix ${x_{0}}\in{\Gamma^{k}}$ and given vertex ${x}$, denote by $|x|$ the number of edges in the shortest path connecting ${x_{0}}$ and $x$. For ${x,y}\in{\Gamma^{k}}$, denote by $d(x,y)$ the number of edges in the shortest path connecting $x$ and $y$. For ${x,y}\in{\Gamma^{k}}$,we write $x\leq y$ if $x$ belongs to the shortest path connecting ${x_{0}}$ with y, and we write $x<y$ if $x\leq y$ and $x\neq y.$ If $x\leq y$ and $|y|=|x|+1$, then we write $x\rightarrow y$. We call vertex ${x_{0}}$ the root of the Cayley tree ${x,y}\in{\Gamma^{k}}$

We set

[TABLE]

[TABLE]

The set $S(x)$ is called the set of direct successors of the vertex $x$.

3. Construction of $p$-adic Gibbs measures for the Ising model

We consider $p$-adic Ising model on the Cayley tree ${\Gamma^{k}}$. Let $\mathbb{Q}_{p}$ be a field of $p$-adic numbers and $\Phi=\{-1,1\}$. A configuration $\sigma$ on V is define by the function $x\in V\to\sigma(x)\in\Phi$. Similarly one can define the configuration ${\sigma_{n}}$ and ${\sigma^{{}_{(n)}}}$ on ${V_{n}}$ and ${W_{n}}$, respectively. The set of all configurations on V (resp.${V_{n}}$,${W_{n}}$ ) is denoted by $\Omega={\Phi^{V}}$ (resp. ${\Omega_{{V_{n}}}}={\Phi^{{V_{n}}}}$,${\Omega_{{W_{n}}}}={\Phi^{{W_{n}}}}$ ).

For given configurations ${\sigma_{n-1}}\in{\Omega_{{V_{n-1}}}}$ and ${\varphi^{(n)}}\in{\Omega_{{W_{n}}}}$ we define a configuration in $\Omega_{{V_{n}}}$ as follows

[TABLE]

A formal $p$-adic Hamiltonian $H:{\Omega}\to\mathbb{Q}_{p}$ of Ising model is defined as

[TABLE]

where $|J|_{p}<p^{-1/(p-1)}$ for any $<x,y>\in L$.

Let $h:x\to{h_{x}}\in\mathbb{Q}_{p}\setminus\{0\}$ be a $p$-adic function on $V$. Consider $p$-adic probability distribution $\mu_{h}^{(n)}$ on ${\Omega_{{V_{n}}}}$, which is defined as

[TABLE]

where ${Z_{n,h}}$ is the normalizing constant

[TABLE]

[TABLE]

A $p$-adic probability distribution $\mu_{h}^{(n)}$ is said to be consistent if for all $n\geq 1$ and ${\sigma_{n-1}}\in{\Omega_{{V_{n-1}}}}$, we have

[TABLE]

In this case, by the $p$-adic analogue of Kolmogorov theorem [4], there exists a unique measure ${\mu_{h}}$ on the set $\Omega$ such that ${\mu_{h}}\left({\left\{{\sigma{|_{{V_{n}}}}\equiv{\sigma_{n}}}\right\}}\right)=\mu_{h}^{(n-1)}\,\,({\sigma_{n-1}})$ for all $n$ and ${\sigma_{n-1}}\in{\Omega_{{V_{n-1}}}}.$

The measure ${\mu_{h}}$ is called $p$-adic generalized Gibbs measure corresponding to the function $h:x\to{h_{x}}\in\mathbb{Q}_{p}$ if restriction of $\mu_{h}$ to $V_{n}$ is a measure (3.2). We notice that if $h_{x}\in\mathcal{E}_{p}$ for all $x\in V$ then corresponding measure is called $p$-adic Gibbs measure. It is said that a phase transition occurs for a given hamiltonian if there exist at least two measures. Moreover, if one of them is not bounded and another one is bounded then it is said that there exists the strong phase transition for that model.

Proposition 3.1**.**

[8]** A sequence of $p$-adic probability distributions $\mu_{h}^{(n)},n=1,2,...$ determined by formula (3.2) is consistent if and only if for any $x\in V\backslash\{{x_{0}}\}$, we have the equality

[TABLE]

where ${{\theta}={{\exp}_{p}}\{2{J}\}}$.

Remark 3.2**.**

It is easy to see that if the function ${h_{x}}$ is a solution to equation (3.5), then the function ${-h_{x}}$ is also a solution. If we consider Ising model on the Cayley tree of order $k$, then these solutions define the same measure ${\mu_{h}}$.

4. $p$-Adic Translation-Invariant Generalized Gibbs Measure

We consider $p$-adic translation-invariant generalized Gibbs measures for Ising model on the Cayley tree ${\Gamma^{3}}$. Thanks to Proposition 4, in order to find all translation-invariant measures it is enough to consider the following equation

[TABLE]

Denoting $z={h^{2}}$, from the last one we get

[TABLE]

which is equivalent to

[TABLE]

The equation (4.3) has at least two solutions $z_{1,2}=\pm 1$. If there exists $\sqrt{\theta^{2}-4}$ in $\mathbb{Q}_{p}$ then Eq. (4.3) has exactly four solutions, which are $z_{1,2}$ and

[TABLE]

Denote $\Delta(\theta)=\theta^{2}-4$.

Lemma 4.1**.**

A number $\sqrt{\Delta(\theta)}$ exists in $\mathbb{Q}_{p}$ if and only if $p\equiv 1(\operatorname{mod}6)$.

Proof.

Since $\theta\in\mathcal{E}_{p}$, due to Lemma 2.4 we have $\left|\theta^{2}-1\right|_{p}<1$. it yields that $|\Delta(\theta)+3|_{p}<1$. Then according to Theorem 2.2, a number $\sqrt{\Delta(\theta)}$ exists in $\mathbb{Q}_{p}$ if and only if $\sqrt{-3}\in\mathbb{Q}_{p}$. Again thanks to Theorem 2.2 existence of $\sqrt{-3}$ is equivalent to the solvability of $x^{2}+3\equiv 0(\operatorname{mod}p)$. It is known that (see [30]) the congruence $x^{2}+3\equiv 0(\operatorname{mod}p)$ is solvable if and only if $p\equiv 1(\operatorname{mod}6)$. ∎

Since $z_{3}z_{4}=1$ one can conclude that the existence of $\sqrt{z_{3}}$ implies the existence of $\sqrt{z_{4}}$ in $\mathbb{Q}_{p}$. For this reason it is enough to check the existence $\sqrt{z_{3}}$ in to describe the set of all $p$-adic translation-invariant generalized Gibbs measures for homogenous Ising model.

Lemma 4.2**.**

The number $\sqrt{z_{3}}$ exists in $\mathbb{Q}_{p}$ if and only if $p\equiv 1(\operatorname{mod}12)$.

Proof.

Assume that $z_{3}$ is a solution of (4.3). Then due to Lemma 4.1 we have $p\equiv 1(\operatorname{mod}6)$. We obtain

[TABLE]

Since $p\neq 2$ and $|\theta-1|_{p}<1$ one has $|z_{3}+1|_{p}<1$. Again keeping in mind $p\neq 2$ and thanks to Theorem 2.2 the existence of $\sqrt{{z_{3}}}$ is equivalent to the existence of the number $\sqrt{-1}$. Then using Corollary 2.3 we conclude that $\sqrt{z_{3}}\in\mathbb{Q}_{p}$ if and only if $p\equiv 1(\operatorname{mod}4)$. After combining $p\equiv 1(\operatorname{mod}6)$ with $p\equiv 1(\operatorname{mod}4)$ we get $p\equiv 1(\operatorname{mod}12)$, which is necessarily and sufficiently condition of the existence $\sqrt{z_{3}}$ in $\mathbb{Q}_{p}$. ∎

Proposition 4.3**.**

Let $\mathcal{N}_{p}$ be a number of solutions of (4.1). Then

[TABLE]

Proof.

Assume that $p\not\equiv 1(\operatorname{mod}4)$. Then due to Corollary 2.3 we have $\sqrt{z_{2}}\not\in\mathbb{Q}_{p}$. Moreover, according to Lemma 4.2 one has $\sqrt{z_{3}},\sqrt{z_{4}}\not\in\mathbb{Q}_{p}$. Hence, in this case (4.1) has exactly two solutions: $h_{1}=\sqrt{z_{1}}$ and $-h_{1}$. Now, we suppose that $p\equiv 1(\operatorname{mod}4)$ and $p\not\equiv 1(\operatorname{mod}3)$. Then again due to Corollary 2.3 there exists $\sqrt{z_{2}}$ in $\mathbb{Q}_{p}$ and by Lemma 4.2 we get $z_{3,4}\not\in\mathbb{Q}_{p}$. Consequently, (4.1) has exactly four solutions: $\pm h_{1}$, $h_{2}=\sqrt{z_{2}}$ and $-h_{2}$. Let us assume that $p\equiv 1(\operatorname{mod}12)$. In this case according to Corollary 2.3 and by Lemma 4.2 there exist $\sqrt{z_{2}}$ and $\sqrt{z_{3,4}}$ in $\mathbb{Q}_{p}$, which imply that Eq. (4.1) has exactly eight solutions: $\pm h_{1},\pm h_{2}$, $h_{3}=\sqrt{z_{3}}$ and $-h_{3}$. ∎

We denote by $TIpGGM(H)$ the set of all $p$-adic translation-invariant generalized Gibbs measures for hamiltonian $H$. Notation $|A|$ means cardinality of the set $A$.

Theorem 4.4**.**

Let $H$ be an Ising model on a Cayley tree order three. Then it holds the following:

[TABLE]

The proof follows from Proposition 4, Remark 3.2 and Proposition 4.3.

Now we study the boundedness of $p$-adic translation-invariant generalized Gibbs measures $\mu_{h_{i}}$, $i=1,2,3,4$. We need some auxiliary lemmas.

Lemma 4.5**.**

[8]** Let $h$ be translation-invariant solution to equation (4.1) and ${\mu_{h}}$ be a corresponding $p$-adic translation-invariant generalized Gibbs measure. Then for normalizing constant ${Z_{n,h}}$ holds

[TABLE]

where

[TABLE]

Lemma 4.6**.**

The norms of the solutions ${h_{1}}$, $h_{2}$, $h_{3}$, $h_{4}$ are equal to one.

Proof.

Since $|z_{i}|_{p}=1$, $i=1,2,3,4$ one can immediately calculate $|\sqrt{z_{i}}|_{p}=1$, $i=1,2,3,4$. ∎

Lemma 4.7**.**

For the normalizing constants ${Z_{n+1,{h_{i}}}}$, $i=1,2,3,4$ we have

1. (i)

\left|Z_{n+1,h_{1}}\right|_{p}=\left\{\begin{array}[]{ll}1,&\mbox{if}\ p\neq 2,\\ 2^{-2^{n}+2},&\mbox{if}\ p=2.\end{array}\right. 2. (ii)

$\left|Z_{n+1,h_{i}}\right|_{p}\leq|\theta-1|_{p}^{2^{n}-2}$, $i=2,3,4$.

Proof.

(i) For normalizing constant ${Z_{n,{h_{1}}}}$ from (4.4) we obtain

[TABLE]

Since $\theta\in\mathcal{E}_{p}$ we have {\left|{{Z_{n+1,{h_{1}}}}}\right|_{p}}=\left\{\begin{array}[]{ll}1,&\mbox{if}\ p\neq 2,\\ {2^{-{2^{n}}+2}},&\mbox{if}\ p=2.\end{array}\right.

(ii) Let ${h_{2}}$ be a solution to (4.1). Then due to non-Archimedean norm’s property we have

[TABLE]

Thus we have ${\left|{{Z_{n+1,{h_{2}}}}}\right|_{p}}=\left|{\theta-1}\right|_{p}^{{2^{n}}-2}.$

(iii) Let ${h_{3}}$ be a solution to (4.1).

[TABLE]

Then using strong triangle inequality we have

[TABLE]

Again noting $\theta\in\mathcal{E}_{p}$ and ${\left|{{h_{3}}}\right|_{p}}=1$ (see Lemma 4.7) we can find

[TABLE]

Using ${z_{3}}{z_{4}}=1$ we obtain the following

[TABLE]

Consequently, ${\left|{{Z_{n+1,{h_{4}}}}}\right|_{p}}\leq\left|{\theta-1}\right|_{p}^{{2^{n}}-2}.$

∎

Theorem 4.8**.**

Let $H$ be an Ising model on a Cayley tree order three. Then the following statements are hold:

1. (i)

if $p=2$ then the unique $p$-adic translation-invariant generalized Gibbs measure ${\mu_{{h_{1}}}}$ is not bounded. 2. (ii)

If $p\neq 2$ then among the $p$-adic translation-invariant generalized Gibbs measures only ${\mu_{{h_{1}}}}$ is bounded.

Proof.

(i) Let $p=2$. In this case due to Theorem 4.4 there exists a unique $p$-adic translation-invariant generalized Gibbs measure ${\mu_{{h_{1}}}}$. Then by Lemma 4.6 and Lemma 4.7 we have

[TABLE]

Hence, $|\mu_{{h_{1}}}^{(n)}(\sigma){|_{2}}\to\infty$ as $n\to\infty$. It means that the measure ${\mu_{{h_{1}}}}$ is not bounded.

(ii) Let $p\neq 2$. According to Lemma 4.6 and Lemma 4.7 one has

[TABLE]

which implies boundedness of limiting measure $\mu_{h_{1}}$. Moreover, if there exists at least one of the measures $\mu_{h_{2}}$, $\mu_{h_{3}}$ and $\mu_{h_{4}}$ then again using Lemma 4.6 and Lemma 4.7 we can verify that they are not bounded. ∎

Thanks to Theorem 4.4 and Theorem 4.8 we get the following

Theorem 4.9**.**

If $p\equiv 1(\operatorname{mod}12)$ then for the Ising model on a Cayley tree order three the strong phase transition occurs.

5. Existence of phase transition for $p$-adic Ising model on $\Gamma^{k}$: $k\geq 3$

5.1. $p$-Adic ART Generalized Gibbs measures

In [8] $p$-adic translation-invariant and periodic generalized Gibbs measures for the Ising model on the Cayley tree of order two are studied. In the previous section we have shown that the strong phase transition occurs on a Cayley tree of order three. In this section we are going to describe new $p$-adic generalized Gibbs measures of the Ising model on the Cayley tree of order $k\geq 3$ by method ART (see [27]).

We recall that each solution of (3.5) define a $p$-adic generalized Gibbs measures for Ising model on the Cayley tree of order $k\geq 1$. One can see that $h_{x}=1$ for all $x\in V$ is a solution of (3.5) for any $k\geq 1$. Now we construct new solutions of (3.5) for $k\geq 3$. If $k=2$ than all translation-invariant solutions of (3.5) can be found from the following

[TABLE]

In [8] have been proved, that the equation (5.1) has a unique solution if $p\not\equiv 1(\operatorname{mod}4)$ and it has exactly three solution if $p\equiv 1(\operatorname{mod}4)$. In what follows we assume that $p\equiv 1(\operatorname{mod}4)$. In this case due to results of [8] the followings

[TABLE]

are solutions of (5.1). For $k\geq 3$ we give construction of some solutions of (3.5) using (5.2).

Let $V^{k}$ be the set of all vertices of the Cayley tree $\Gamma^{k}$. Since $k>2$ one can consider $V^{2}$ as a subset of $V^{k}$. Define the following function

[TABLE]

where $i=1,2$. This function on the Cayley tree of order $k=3$ is shown in Fig.1.

Now we shall check that (5.3) satisfies (3.5) on $\Gamma^{k}$.

Let $x\in V^{2}\subset V^{k}$. For $i=1,2$ we have

[TABLE]

here we used

[TABLE]

If $x\in V^{k}\setminus V^{2}$ then it is easy to see that $S_{k}(x)\subset V^{k}\setminus V^{2}.$ Therefore we have

[TABLE]

Thus $\widetilde{h}^{{(i)}}_{x},i=1,2$ satisfies the functional equation (3.5) and we denote by $\mu_{\widetilde{h}^{{(i)}}_{x}},i=1,2$ the Gibbs measures corresponding to $\widetilde{h}^{{(i)}}_{x},i=1,2$ and those measures we called $p$-adic ART generalized Gibbs measures. Thus, we have the following result:

Theorem 5.2**.**

Let $p\equiv 1(\operatorname{mod}4)$. Then there exists at least three $p$-adic generalized Gibbs measures for the Ising model on a Cayley tree order $k\geq 3$.

Aknowledgements The authors thank Professor U.A.Rozikov for useful discussions.