$p$-adic boundary laws and Markov chains on trees

TL;DR

This paper investigates $p$-adic boundary laws and Markov chains on infinite trees, establishing conditions for uniqueness, existence of multiple chains, and boundedness properties using adapted boundary law methods in the $p$-adic setting.

Contribution

It introduces a systematic adaptation of boundary law techniques to the $p$-adic context, revealing new phenomena in Markov chains on trees.

Findings

Uniqueness of $p$-adic Markov chains under certain conditions.

Existence of multiple $p$-adic Markov chains for specific stochastic matrices.

Boundedness of chains depending on the $p$-adic norm relations.

Abstract

In this paper we consider $q$-state potential on general infinite trees with a nearest-neighbor $p$-adic interactions given by a stochastic matrix. {We show the uniqueness of the associated Markov chain ({\em splitting Gibbs measures}) under some sufficient conditions on the stochastic matrix.} Moreover, we find a family of stochastic matrices for which there are at least two $p$-adic Markov chains on an infinite tree (in particular, on a Cayley tree). When the $p$-adic norm of $q$ is greater ({{\em resp.}} less) than the norm of any element of the stochastic matrix then it is proved that the $p$-adic Markov chain is bounded ({{\em resp.}} is not bounded). Our method {uses} a classical boundary law argument carefully adapted from the real case to the $p$-adic case, by a systematic use of some nice peculiarities of the ultrametric ($p$-adic) norms.

Equations

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

$$|x+y{|}_p\le \max \{|x{|}_p,|y{|}_p\}.$$

$$\mathrm{If}|x{|}_p\ne |y{|}_p\mathrm{then}|x+y{|}_p=\max \{|x{|}_p,|y{|}_p\}.$$

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

$$\begin{array}[]{lll}F(a_{0})\equiv 0\,(\operatorname{mod}p^{2\gamma+1}),\\[5.69054pt] F^{\prime}(a_{0})\equiv 0\,(\operatorname{mod}p^{\gamma}),\\[5.69054pt] F^{\prime}(a_{0})\neq 0\,(\operatorname{mod}p^{\gamma+1}).\end{array}$$

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

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

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

$$|{\exp }_p(x){|}_p=1,|{\exp }_p(x)-1{|}_p=|x{|}_p,|{\log }_p(1+x){|}_p=|x{|}_p,$$

$${\log }_p({\exp }_p(x))=x,{\exp }_p({\log }_p(1+x))=1+x.$$

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

$$L^z=\{⟨x,y⟩\in L:d(z,x)=d(z,y)+1\},$$

$$\text{prescript}zL=\{⟨x,y⟩\in L:d(z,y)=d(z,x)+1\}.$$

$$\partial A=\{x\in V\setminus A:\exists y\in A,⟨x,y⟩\}.$$

$${\alpha }_{i,x}{P}_b(i,j)={\alpha }_{j,y}{P}_b(j,i),\forall i,j\in \Phi .$$

$$\mu ({\sigma }_{\Lambda }=\zeta )={\alpha }_z({\zeta }_z)\prod _{\genfrac{}{}{0ex}{}{⟨x,y⟩{\in }^{z}L:}{x,y\in \Lambda }}{P}_{⟨x,y⟩}({\zeta }_x,{\zeta }_y).$$

$$\displaystyle\begin{array}[]{lll}Q_{\langle x,y\rangle}(i,j)=Q_{\langle y,x\rangle}(j,i),\\[8.53581pt] \sum_{j=1}^{q}Q_{\langle x,y\rangle}(j,i)=1.\end{array}$$

$$z_i(x,y)=c(x,y)\prod _{v\in \partial \{x\}\setminus \{y\}}\sum _{j\in \Phi }{z}_j(v,x)Q_{⟨v,x⟩}(j,i),$$

$$\mathbf{z}=\{\mathbf{z}(x,y)=(z_1(x,y),\dots ,z_q(x,y)){\}}_{⟨x,y⟩\in L}$$

$$\mu^{\boldsymbol{z}}(\sigma_{\mbox{$\;\!\!$}\bar{\varLambda}}\mbox{$\;\!\!$}=\varsigma)=\frac{1}{Z_{\mbox{$\;\!\!$}\bar{\varLambda}}}\prod_{x\in\partial\Lambda}{z}_{\varsigma(x)}(x,x_{\Lambda})\prod_{{b\in L:\atop b\cap\Lambda\neq\emptyset}}Q_{b}(\varsigma_{b}),$$

$${\mathcal{E}}_p=\left\{x\in {\mathbb{Q}}_p:|x-1{|}_p\le \frac{1}{p}\right\}.$$

$$|a_i{|}_p=1,|a_i-1{|}_p\le M,$$

$${\left|\prod _{i=1}^m{a}_i\right|}_p=1,{\left|\prod _{i=1}^m{a}_i-1\right|}_p\le M.$$

$$z_i(x,y)=\prod _{v\in \partial \{x\}\setminus \{y\}}\frac{1+{\sum }_{j=1}^{q-1}(z_j(v,x)-1)Q_{⟨v,x⟩}(j,i)}{1+{\sum }_{j=1}^{q-1}(z_j(v,x)-1)Q_{⟨v,x⟩}(j,q)},i=1,2,\dots ,q-1.$$

$$Q_{⟨v,x⟩}(q,i)=1-\sum _{j=1}^{q-1}{Q}_{⟨v,x⟩}(j,i),i=1,2,\dots ,q.$$

$$\begin{array}[]{lll}|Q_{\langle x,y\rangle}(j,i)|_{p}\leq 1,\\[8.53581pt] \left|Q_{\langle x,y\rangle}(j,i)-Q_{\langle x,y\rangle}(j,q)\right|_{p}\leq{1\over p},\ \ \forall\langle x,y\rangle,\ \ \forall i,j.\end{array}$$

$$\sum _{j=1}^q{Q}_{⟨v,x⟩}(j,i)=1,\forall ⟨v,x⟩,\forall i,$$

$$\Vert z\Vert =\underset{i}{\max }|z_i{|}_p.$$

$$K_i\equiv K_i(v,x,q)=\frac{1+{\sum }_{j=1}^{q-1}(z_j(v,x)-1)Q_{⟨v,x⟩}(j,i)}{1+{\sum }_{j=1}^{q-1}(z_j(v,x)-1)Q_{⟨v,x⟩}(j,q)}.$$

$$|K_i{|}_p={\left|\frac{1+{\sum }_{j=1}^{q-1}(z_j(v,x)-1)Q_{⟨v,x⟩}(j,i)}{1+{\sum }_{j=1}^{q-1}(z_j(v,x)-1)Q_{⟨v,x⟩}(j,q)}\right|}_p=1.$$

Full text

Mathematics Subject Classifications (2010). 46S10, 82B26, 12J12 (primary); 60K35 (secondary)

Key words. Cayley trees, boundary laws, Gibbs measures, translation invariant measures, $p$-adic numbers, $p$-adic probability measures, $p$-adic Markov chain, non-Archimedean probability.

1. Introduction

In this paper we develop a boundary law argument to study $p$-adic Markov chains on general trees. In the real case Markov chains on trees are particular cases of Gibbs measures corresponding to a Hamiltonian with nearest-neighbor interactions. In the theory of Gibbs measures on trees (see [8, Chapter 12] and [23]) the main problem is to describe the set of limiting Gibbs measures corresponding to a given Hamiltonian. A complete analysis of this set is often a difficult problem, this is even not completely described for the Ising model (see [3, 4, 5, 25] for some recent results).

Parallel to the real valued Gibbs measures, the $p$-adic Gibbs measures are studied using the $p$-adic mathematical physics in [2, 12, 13, 24, 29]. A $p$-adic distribution is an analogue of ordinary distributions that takes values in a ring of $p$-adic numbers [11], [12]. Analogically to a measure on a measurable space, a $p$-adic measure is a special case of a $p$-adic distribution. A $p$-adic distribution taking values in a normed space is called a $p$-adic measure if the values on compact open subsets are bounded.

It is known that some $p$-adic models in physics cannot be described using ordinary Kolmogorov’s probability theory [13, 15, 17, 29]. In [14] the $p$-adic probability theory was developed using the theory of non-Archimedean measures [21]. In [6, 10, 18, 19, 20, 26] various models of statistical physics in the context of $p$-adic fields are studied.

In probability theory Kolmogorov’s extension theorem (see, e.g., [28, Chapter II, § 3, Theorem 4, page 167]), says that a compatibility condition of a sequence of probability measures ensures that there exists a unique (limit) measure. This theorem is used to introduce (real-valued) Markov chains on trees (see [8, Chapter 12]) by notion of a boundary law. A $p$-adic analogue of Kolmogorov’s theorem was proved in [7]. Such a $p$-adic Kolmogorov theorem allows us to construct wide classes of stochastic processes and to develop statistical mechanics in the context of $p$-adic theory [16]-[20].

In the present paper we introduce $p$-adic Markov chains on general infinite trees. Such chains are constructed by $p$-adic boundary laws (for the real case see [8, Chapter 12]). We also discuss the uniqueness and boundedness of the $p$-adic Markov chain. The boundedness of the $p$-adic measure is needed to integrate $p$-adic valued functions [11, 12, 27], and also to consider conditional expectations [11, 16]. Note that $p$-adic measures are also useful in $p$-adic $L$-functions following the works of B. Mazur (see [9, 15] for details).

The paper is organized as follows. Section 2 presents definitions and known results. Section 3 is devoted to an introduction of $p$-adic Markov chains through boundary laws. Section 4 (resp. Section 5) is devoted to finding a sufficient condition of the uniqueness (resp. non-uniqueness) of $p$-adic Markov chain. In Section 6 we give some conditions ensuring that the $p$-adic Markov chain is (resp. not) bounded.

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 $x=p^{r}{n\over m}$, where $r,n\in\mathbb{Z}$, $m$ is a positive integer, and $n$ and $m$ are relatively prime with $p$: $(p,n)=1$, $(p,m)=1$. The $p$-adic norm of $x$ is given by

[TABLE]

This norm is non-Archimedean and satisfies the so-called strong triangle inequality

[TABLE]

We will often use the following fact:

[TABLE]

The completion of $\mathbb{Q}$ with respect to the $p$-adic norm defines the $p$-adic field $\mathbb{Q}_{p}$. Any $p$-adic number $x\neq 0$ can be uniquely represented in the canonical form

[TABLE]

where $\gamma(x)\in\mathbb{Z}$ and the integers $x_{j}$ satisfy: $x_{0}>0$, $0\leq x_{j}\leq p-1$ (see [15, 27, 29]). In this case $|x|_{p}=p^{-\gamma(x)}$.

Our analysis will strongly relies on nice properties of the $p$-adic norm, and on the two following classical results in $p$-adic algebra.

Theorem 1** ([15, 29]).**

The equation $x^{2}=a$, $0\neq a=p^{\gamma(a)}(a_{0}+a_{1}p+...),0\leq a_{j}\leq p-1$, $a_{0}>0$ has a solution $x\in\mathbb{Q}_{p}$ if and only if the following conditions are fulfilled:

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

ii) $a_{0}$ is a quadratic residue modulo $p$ if $p\neq 2$; $a_{1}=a_{2}=0$ if $p=2$.

The elements of the set $\mathbb{Z}_{p}=\{x\in\mathbb{Q}_{p}:|x|_{p}\leq 1\}$ are called $p$-adic integers.

The following statement is known as Hensel’s lemma [1, Theorem 3.15].

Theorem 2**.**

Let $F(x)=\sum_{i=0}^{n}c_{i}x^{i}$ be a polynomial whose coefficients are $p$-adic integers. Let $F^{\prime}(x)=\sum_{i=0}^{n}ic_{i}x^{i-1}$ be the derivative of $F(x)$. Assume there exist $a_{0}\in\mathbb{Z}_{p}$ and $\gamma\in\{0,1,2,\dots\}$ such that

[TABLE]

Then there exists $a\in\mathbb{Z}_{p}$ such that $F(a)=0$ and $a\equiv a_{0}\,(\operatorname{mod}p^{\gamma+1})$.

Given $a\in\mathbb{Q}_{p}$ and $r>0$ put

[TABLE]

The $p$-adic logarithm is defined by the series

[TABLE]

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

[TABLE]

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

Lemma 1** ([15]).**

Let $x\in B(0,p^{-1/(p-1)})$, then

[TABLE]

[TABLE]

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

[TABLE]

A $p$-adic measure is called a $p$-adic probability measure if $\mu(X)=1$, see, e.g. [11, 21]. Let us warn that due to the different axiomatic and ring of values, some intuitive properties of sets of probability measures (like e.g. some convex properties) are not valid anymore [24].

2.2. Tree.

A tree is a connected graph without cycles (see [22] for more details). Let $\mathcal{T}=(V,L)$ be a tree, where $V$ is the set of vertices and $L$ is the set of edges. Two vertices $x$ and $y$ are called nearest neighbors if there exists an edge $b\in L$ connecting them. We will use the notation $b=\langle x,y\rangle$ for the edge connecting the vertices $x$ and $y$. A collection of nearest neighbor pairs $\langle x,x_{1}\rangle,\langle x_{1},x_{2}\rangle,...,\langle x_{d-1},y\rangle$ is called a path from $x$ to $y$. The distance $d(x,y)$ on the tree is the number of edges of the shortest path from $x$ to $y$.

For $z\in V$, we denote

[TABLE]

[TABLE]

Let $A\subset V$. Denote

[TABLE]

3. $p$-adic Markov chain and boundary laws

We consider a system with nearest neighbor interactions on a tree where the spins assigned to the vertices of the tree take values in the set $\Phi:=\{1,2,\dots,q\}$.

A configuration $\sigma_{A}$ on $A\subset V$ is then defined as a function $x\in A\mapsto\sigma_{A}(x)\in\Phi$. The set of all configurations is $\Phi^{A}$.

By $p$-adic probability vector we mean a vector with $p$-adic valued coordinates summing to 1. A $p$-adic stochastic matrix is a matrix with each row being a $p$-adic probability vector.

For each edge $b=\langle x,y\rangle\in L$ we consider a stochastic matrix $\mathbb{P}_{b}=\left(P_{b}(i,j)\right)_{i,j=1}^{q}$. For each $x\in V$ consider a probability vector $\alpha_{x}=(\alpha_{1,x},\dots,\alpha_{q,x})$.

For any edge $b=\langle x,y\rangle\in L$ we assume that

[TABLE]

Definition 1**.**

A $p$-adic probability distribution (measure) $\mu$ is called a $p$-adic Markov chain with transition matrices $(P_{b})_{b\in L}$ and marginal distribution $\alpha_{x}$ at $x\in V$ if for all finite, connected set $\Lambda\subset V$, and all $\zeta\in\Phi^{\Lambda}$ and $z\in\Lambda$ the following holds

[TABLE]

Note that the reversibility condition (3.1) is equivalent to the statement that the expression on the right of (3.2) is independent of the choice of $z\in\Lambda$.

Consider for each edge $b=\langle x,y\rangle$ a matrix $\mathcal{Q}_{b}=(Q_{b}(i,j))_{i,j=1}^{q}$. We always assume

[TABLE]

Let $\boldsymbol{z}(x,y)=(z_{1}(x,y),\dots,z_{q}(x,y))$ be a vector in $\mathbb{Q}_{p}$.

Definition 2**.**

For $(\mathcal{Q}_{b})_{b\in L}$ satisfying (3.5), a $p$-adic boundary law111Compare with real boundary law of [8, Definition (12.10)]. $\{\boldsymbol{z}(x,y)\}_{\langle x,y\rangle\in L}$ is such that for any $\langle x,y\rangle\in L$, and for all $i\in\Phi$, it holds

[TABLE]

where $c(x,y)$ is an arbitrary constant (not depending on $i\in\Phi$).

Using (3.5) and proceeding as in the classical case of [8, Formula (12.13), page 243], one directly gets that each boundary law

[TABLE]

defines a $p$-adic Markov chain $\mu^{\boldsymbol{z}}$: for any finite connected set $\varnothing\neq\varLambda\subset V$ (and $\bar{\varLambda}=\varLambda\cup\partial\varLambda$), one has

[TABLE]

where $Z_{\bar{\varLambda}}=Z_{\bar{\varLambda}}(\boldsymbol{z})$ is the normalizing factor, x_{\mbox{;!!}\varLambda} denotes the unique neighbor of $x\in\partial\varLambda$ belonging to $\varLambda$, and $\varsigma_{b}=(\varsigma(u),\varsigma(v))$, for $b=\langle u,v\rangle$. We stress that the first condition in (3.5), which is [8, Formula (12.9)], is needed to check that $\mu^{\boldsymbol{z}}$ is a well defined $p$-adic Markov chain.

4. Criterion for uniqueness of the $p$-adic Markov chain

A $p$-adic Markov chain can be considered as a particular case of $p$-adic Gibbs measure defined through the $p$-adic exponential $\exp_{p}(x)$, with $|x|_{p}<p^{-1/(p-1)}$ [18]. As it was mentioned above, the set of values of a $p$-adic norm $|\cdot|_{p}$ is $\{p^{m}:m\in\mathbb{Z}\}$, so the condition $|x|_{p}<p^{-1/(p-1)}$ is equivalent to the condition $|x|_{p}\leq{1\over p}$. Consequently, we shall restrict part of the analysis to quantities belonging to the set:

[TABLE]

The following lemma will also be useful (see [18, Lemma 4.6]).

Lemma 2**.**

If $a_{i}\in\mathbb{Q}_{p}$ for all $i=1,\dots,m$ are such that

[TABLE]

then

[TABLE]

Without loss of generality, we set hereafter $z_{q}(v,x)\equiv 1$ (a normalization at $q$). Then the condition (3.6) for the stochastic matrix $\mathcal{Q}_{b}=(Q_{b}(i,j))_{i,j=1}^{q}$ reads

[TABLE]

Here we have used

[TABLE]

In this section we examine the conditions on the parameters $k\geq 1$ and on $\mathcal{Q}_{b}$ for the existence and the uniqueness of the solutions of the equation (4.1).

For the uniqueness, we ssume that the matrix $\mathcal{Q}_{b}=(Q_{b}(i,j))_{i,j=1}^{q}$ satisfies the following conditions

[TABLE]

Theorem 3**.**

Assume each vertex of the tree has degree at least $2$ and that the matrix $\mathcal{Q}_{b}=(Q_{b}(i,j))_{i,j=1}^{q}$ satisfies (3.5) and (4.2). Then the equation (4.1) has a unique solution $\boldsymbol{z}(x,y)\equiv(1,1,\dots,1)\in{\mathcal{E}}^{q-1}_{p}$, $\langle x,y\rangle\in L$.

Proof.

Since

[TABLE]

it follows that $\boldsymbol{z}(x,y)\equiv(1,1,\dots,1)$ is a solution to (4.1).

We show its uniqueness. For $z=(z_{1},\dots,z_{q-1})\in\mathbb{Q}_{p}^{q-1}$, we introduce the norm

[TABLE]

Let $\boldsymbol{z}(x,y)\in{\mathcal{E}}^{q-1}_{p}$, $\langle x,y\rangle\in L$ be a solution. Denote

[TABLE]

Using (2.1), (4.2) and Lemma 2, we calculate $|K_{i}|_{p}$:

[TABLE]

Let us now estimate $|K_{i}-1|_{p}$ using (4.2):

[TABLE]

where we have used the hypothesis

[TABLE]

and $\hat{v}\equiv\hat{v}(x,y)$ is defined by

[TABLE]

Thus $K_{i}$ satisfies the conditions of Lemma 2, and we have

[TABLE]

Consequently,

[TABLE]

Since this estimation is true for arbitrary edge $\langle x,y\rangle\in L$, one can start from any edge and then iterate the estimation (4.5), to obtain the following

[TABLE]

which as $n\to\infty$ gives $\boldsymbol{z}(x,y)\equiv 1.$ ∎

Denote by $\mu^{\boldsymbol{1}}$ the $p$-adic Markov chain which corresponds to $\boldsymbol{z}(x,y)\equiv(1,\dots,1).$

Corollary 1**.**

Under the conditions of Theorem 3, there exists a unique $p$-adic Markov chain, which satisfies that for any finite connected set $\varnothing\neq\varLambda\subset V$ (and $\bar{\varLambda}=\varLambda\cup\partial\varLambda$),

[TABLE]

where

[TABLE]

5. Criterion for non-uniqueness of the $p$-adic Markov chains

5.1. On a regular tree.

Consider the Cayley tree of order $k\geq 1$. Suppose the matrix $\mathcal{Q}_{b}$ in the system of equations (4.1) satisfies the condition

[TABLE]

We assume further that $Q_{\langle v,x\rangle}(1,1)$ and $Q_{\langle v,x\rangle}(1,q)$ are independent on $\langle v,x\rangle$, that is

[TABLE]

Theorem 4**.**

If (5.1), (5.2) are satisfied, $\alpha,\beta$ are $p$-adic integers, and there exists $\gamma\in\{0,1,2,\dots\}$ such that

[TABLE]

then the equation (4.1) has at least two solutions.

Proof.

We shall prove that the equation (4.1) has two constant (translational-invariant) solution $\boldsymbol{z}(x,y)\equiv\boldsymbol{z}$, $\forall\langle x,y\rangle\in L$. The first solution is already known: $\boldsymbol{z}(x,y)\equiv(1,\dots,1)$. We shall show that the system (4.1) has a solution of the following form

[TABLE]

Then from (4.1), for the Cayley tree of order $k\geq 2$, we get

[TABLE]

Independently on parameters, this equation has solution $z=1$. We are going to find conditions on $\alpha\neq\beta$ and on $k$ to have at least one solution $z\neq 1$.

The equation (5.7) can be written as $F(z)=0$ with

[TABLE]

We are interested in the solution of $G(z)={F(z)\over z-1}=0$, where

[TABLE]

Since $\alpha,\beta$ are $p$-adic integers, $G(z)$ has only $p$-adic integer coefficients. Now we shall check the other conditions of Hensel’s lemma (see Theorem 2). Take $a_{0}=1$. Then we have $G(1)=1+k(\beta-\alpha)$ and

[TABLE]

Therefore by (5.6), the conditions of Hensel’s lemma are satisfied for $G(z)$. Hence there exists a $p$-adic integer $a$ such that $G(a)=0$ and $a\equiv a_{0}\,(\operatorname{mod}p^{\gamma+1})$, i.e. $G(z)=0$ has a solution $z=a$. Since $a_{0}=1$, we have $a\equiv 1\,(\operatorname{mod}p^{\gamma+1})$. Thus $a\in{\mathcal{E}}_{p}$. This proves the theorem. ∎

Remark 1**.**

Note that if $p$ divides $k(\beta-\alpha)+1$ then $p$ does not divide $\beta-\alpha$, therefore $|\beta-\alpha|_{p}=1>{1\over p}$, i.e. the condition (4.2) is not satisfied.

Let us give some examples of parameters satisfying the conditions of Theorem 4:

Example 1**.**

The case $\gamma=0$:

• a)

Let $k=1$. Then the equation $G(z)=0$ has a unique solution $z=a={\alpha-1\over\beta}$. The condition (5.6) of Theorem 4 is equivalent to

[TABLE]

This implies $|\alpha-1|_{p}=1$, and $|a|=1$, $|a-1|\leq{1\over p}$. Thus, the solution $z=a$, other than the solution $z=1$, is also in $\mathcal{E}_{p}$.

• b)

Take $k=2$, $p=3$, $\alpha=2$, $\beta=3$. Then $k(\beta-\alpha)+1=3\equiv 0\ (\rm{mod}\ 3)$ and $k\beta+{k(k-1)\over 2}(\beta^{2}-\alpha^{2})=11\not\equiv 0\ (\rm{mod}\ 3)$. For these parameters the equation (5.7) has three solutions:

[TABLE]

Note (see Theorem 1) that $\sqrt{13}$ exists in $\mathbb{Q}_{3}$. Moreover, it can be calculated222http://www.numbertheory.org/php/p-adic.html*:*

[TABLE]

Then we get

[TABLE]

[TABLE]

Hence $z_{1}\in\mathcal{E}_{3}$, and $z_{1}$ plays the role of $a\in\mathcal{E}_{3}$ mentioned in the proof of Theorem 4. On the other hand, we have $z_{1}z_{2}={1\over 9}$. Consequently $|z_{1}z_{2}|_{3}=9$. Since $|z_{1}|_{3}=1$, we obtain $|z_{2}|_{3}=9$. Thus $z_{2}\notin\mathcal{E}_{3}$.

Example 2**.**

The case $\gamma=1$: Take $k=2$, $p=3$, $\alpha=6$, $\beta=19$. Then

[TABLE]

In this case the equation (5.7) has three solutions:

[TABLE]

We have $|z_{1}-1|_{3}=|{363-39\sqrt{61}\over 722}|_{3}\leq{1\over 3}.$ Thus $z_{1}\in\mathcal{E}_{3}$. Similarly, one can see that $z_{2}\in\mathcal{E}_{3}$.

As a corollary of Theorem 4, we have the following.

Theorem 5**.**

If the conditions of Theorem 4 are satisfied then for the matrix $\mathcal{Q}_{b}$ on the Cayley tree of order $k\geq 1$, there are at least two $p$-adic Markov chains.

Remark 2**.**

Theorem 4 can be generalized as follows: fix $m\in\{1,2,\dots,q-1\}$ and assume

[TABLE]

Suppose $\sum_{j=1}^{m}Q_{\langle v,x\rangle}(j,i)$ and $\sum_{j=1}^{m}Q_{\langle v,x\rangle}(j,q)$ are independent on $\langle v,x\rangle$, i.e.,

[TABLE]

Under the above mentioned conditions one can show that the system (4.1) has a solution of the following form

[TABLE]

Then from (4.1), for the Cayley tree of order $k\geq 2$, we get

[TABLE]

This equation is identical with (5.7) and it has non-unique solutions when $A$ and $B$ (replacing $\alpha$ and $\beta$) satisy the conditions mentioned in Theorem 4.

5.2. Extension on a non-regular tree.

Consider now a general tree $\mathcal{T}$, with each vertex having at least two nearest neighbors. Recall that $L$ is the set of all edges of $\mathcal{T}$. Such a tree contains a Cayley tree (of some order $k\geq 1$) as a subtree, which we denote by $\Gamma^{k}$. Let $L_{k}$ be the set of all edges of $\Gamma^{k}$, i.e., $L_{k}\subset L$.

Assume on $\Gamma^{k}$, the conditions of Theorem 4 are satisfied. Then we have a boundary law of the form

[TABLE]

Let $g(z)={1-\alpha+\alpha z\over 1-\beta+\beta z}.$ Define on the edges $\langle x,y\rangle$ of the general tree $\mathcal{T}$ the following vector-valued function

[TABLE]

where

[TABLE]

and $z$ is defined in (5.11).

For $i=2,\dots,q-1$, we assume

[TABLE]

and show that $\boldsymbol{l}$ defined by (5.12) satisfies the equation (4.1).

For coordinates $l_{i}(x,y)=1$, $i=2,3,\dots,q-1$, from (4.1) we have

[TABLE]

Therefore, by (5.1), (5.13) and (5.14), one can see that the right-hand side of (5.15) is always 1.

Now we show that $l_{1}(x,y)$ also satisfies (4.1). Indeed, we note that $\partial\{x\}\setminus\{y\}=A_{k}(x,y)\cup B_{k}(x,y)$, where $A_{k}(x,y)=(\partial\{x\}\setminus\{y\})\cap L_{k}$ and $B_{k}(x,y)=(\partial\{x\}\setminus\{y\})\cap(L\setminus L_{k})$.

We thus have the following three possible cases:

Case: $x,y\in\Gamma^{k}$. In this case $l_{1}(x,y)=z$ and $A_{k}(x,y)$ has $k$ elements. Therefore, the equation (4.1) for $l_{1}(x,y)$ is reduced to $z=(g(z))^{k}$, which is satisfied by the conditions of Theorem 4.

Case: $\langle x,y\rangle\in L,\ \ x\in L\setminus L_{k}$. Then $A_{k}(x,y)=\emptyset$ and hence the equation (4.1) for $l_{1}(x,y)$ is reduced to the identity $1=1$.

Case: $\langle x,y\rangle\in L,\ \ x\in\Gamma^{k},\ \ y\in L\setminus L_{k}$. In this case $A_{k}(x,y)$ contains $k+1$ elements, and we have $l_{1}(v,x)=z$ for all $v\in A_{k}(x,y)$. Thus the equation (4.1) has the form $l_{1}(x,y)=(g(z))^{k+1}$. Using $z=(g(z))^{k}$, we get $l_{1}(x,y)=zg(z)$ as in the definition (5.12). Thus $\boldsymbol{l}(x,y)$ satisfies the equation (4.1).

Denote by $\mu^{\boldsymbol{l}}$ the $p$-adic Markov chain corresponding to $\boldsymbol{l}$ given by (5.12).

We have proved the following theorem.

Theorem 6**.**

Let $\mathcal{T}$ be a tree containing a Cayley tree $\Gamma^{k}$ of order $k\geq 1$, as a subtree. Suppose the conditions of Theorem 4 are satisfied on $\Gamma^{k}$. If (5.14) is satisfied, then on the tree $\mathcal{T}$ there are at least two $p$-adic Markov chains (one is $\mu^{\boldsymbol{l}}$ and the other is $\mu^{\boldsymbol{1}}$).

6. Criterion for the (un-)boundedness of the $p$-adic Markov chains

Now we are interested in finding out whether a $p$-adic Markov chain is bounded.

Let $\{\boldsymbol{z}(x,y)\in\mathcal{E}_{p},\ \ \langle x,y\rangle\in L\}$ be a boundary law for the matrix $\mathcal{Q}_{b}=(Q_{b}(i,j))$ and $\mu^{\boldsymbol{z}}$ be the corresponding $p$-adic Markov chain.

Theorem 7**.**

The following hold

if $\max_{i,j\in\Phi}|Q_{b}(i,j)|_{p}\leq|q|_{p}$ for all $b\in L$, then the $p$-adic Markov chain $\mu^{\boldsymbol{z}}$ is bounded;

• 2)

if $\min_{i}\max_{j}|Q_{b}(i,j)|_{p}>|q|_{p}$ for all $b\in L$, then the $p$-adic Markov chain $\mu^{\boldsymbol{z}}$ is not bounded.

Proof.

It suffices to show that for any finite connected set $\varnothing\neq\varLambda\subset V$ (denote $\bar{\varLambda}=\varLambda\cup\partial\varLambda$), and any $\varsigma\in\Omega_{\Lambda}$, one has |\mu^{\boldsymbol{z}}(\sigma_{\mbox{;!!}\bar{\varLambda}}\mbox{;!!}=\varsigma)|_{p}\leq M, for some $M>0$. Using (3.7), we get

[TABLE]

Let us calculate

[TABLE]

The set $\bar{\varLambda}$ can be decomposed as

[TABLE]

where $\partial_{int}A=\{x\in A:\exists y\in V\setminus A,\ \langle x,y\rangle\}$. Since $\mathcal{Q}_{b}$ is stochastic for any $b\in L$ we get

[TABLE]

1. Under the conditions of the part 1), we have (note that $|q|_{p}\leq 1$)

[TABLE]

Thus

[TABLE]

2. Suppose now the conditions of part 2) are satisfied. For a marginal on the two-site volume, i.e., an edge $b=\langle x,y\rangle$, corresponding to a boundary law $\boldsymbol{z}=(z_{1}(x,y),\dots,z_{q}(x,y))$, when $\sigma(x)=i$ is fixed we have

[TABLE]

Therefore,

[TABLE]

In order to show that the measure $\mu^{\boldsymbol{z}}$ is not bounded, it is enough to show that its marginal measure is not bounded. Let $\pi=\{x_{0},x_{1},...\}$ be an arbitrary infinite path in the tree. The marginal measure $\mu^{\boldsymbol{z}}_{\pi}$ has the form

[TABLE]

Here $\omega_{n}:\{x_{0},...,x_{n}\}\to\Phi=\{1,2,\dots,q\}$ is a configuration on $\{x_{0},...,x_{n}\}$ and $\alpha_{i}$ is a coordinate of the invariant stochastic vector of the matrix $\left(\mu_{\langle x_{0},x_{1}\rangle}^{\boldsymbol{z}}(i,j)\right)_{i,j=1,\dots,q}$.

To ensure that $|\alpha_{\omega_{n}(x_{0})}|_{p}>c$ for some $c>0$. We can choose the value $i_{0}=\omega_{n}(x_{0})$ (of the configuration $\omega_{n}$ on the vertex $x_{0}$) such that

[TABLE]

Then since $\alpha$ is a probability vector we have

[TABLE]

Having $i_{0}$, we choose the value $i_{1}=\omega_{n}(x_{1})$ of the configuration $\omega_{n}$ to satisfy

[TABLE]

By iterating, we define $i_{m}=\omega(x_{m})$ to have

[TABLE]

Then for the above constructed $\omega_{n}$, by (6.3) we get

[TABLE]

Here, at the last step we have used the following (which is true by the condition of the part 2) of theorem)

[TABLE]

Consequently, for such a configuration $\omega_{n}$, from (6.4) and (6.5), we find that

[TABLE]

i.e., $\mu^{\boldsymbol{z}}$ is not bounded. ∎

Acknowledgements

UAR thanks the University Paris-Est Créteil (UPEC) for the hospitality during June 2019, where this work has been achieved, and Labex Bézout (Université Paris Est) for the financial and logistic support of this visit. The collaboration of the authors is realized within the project "Real/ $p$-adic dynamical systems and Gibbs measures" funded by LabEx Bézout (ANR-10-LABX-58).