Phase transition for the once-excited random walk on general trees

TL;DR

This paper investigates the phase transition and recurrence properties of once-excited and multi-excited random walks on general trees, extending previous work on $M$-digging random walks and exploring conditions for recurrence and transience.

Contribution

It analyzes the critical behavior of once-excited random walks on superperiodic trees and establishes transience results for multi-excited walks on trees with branching number greater than one.

Findings

Critical $M$-digging random walk on superperiodic trees is recurrent.

Multi-excited random walk on trees with branching number > 1 is transient.

Techniques from previous studies are effectively applied to new classes of excited random walks.

Abstract

The phase transition of $M$-digging random on a general tree was studied by Collevecchio, Huynh and Kious (2018). In this paper, we study particularly the critical $M$-digging random walk on a superperiodic tree that is proved to be recurrent. We keep using the techniques introduced by Collevecchio, Kious and Sidoravicius (2017) with the aim of investigating the phase transition of Once-excited random walk on general trees. In addition, we prove if $\mathcal T$ is a tree whose branching number is larger than $1$, any multi-excited random walk on $\mathcal{T}$ moving, after excitation, like a simple random walk is transient.

Figures

Equations

$$\alpha (\mathcal{T},{\lambda }_1,\lambda )=\underset{n\to \infty }{\mathrm{liminf}}{\left(\prod _{i=1}^n\frac{{\lambda }^2+(x_i-1){\lambda }_1\lambda +{\lambda }_1}{1+x_i{\lambda }_1}\right)}^{1/n}.$$

$$\beta (\mathcal{T},{\lambda }_1,\lambda )=\underset{n\to \infty }{\mathrm{limsup}}{\left(\prod _{i=1}^n\frac{{\lambda }^2+(x_i-1){\lambda }_1\lambda +{\lambda }_1}{1+x_i{\lambda }_1}\right)}^{1/n}.$$

$$\gamma (\mathcal{T},{\lambda }_1)=\underset{n\to \infty }{\mathrm{liminf}}\frac{-{\sum }_{i=1}^n\ln \left[1-\frac{(x_i-1){\lambda }_1+2}{(1+x_i{\lambda }_1)i}\right]}{\ln n}.$$

$$\eta (\mathcal{T},{\lambda }_1)=\underset{n\to \infty }{\mathrm{limsup}}\frac{-{\sum }_{i=1}^n\ln \left[1-\frac{(x_i-1){\lambda }_1+2}{(1+x_i{\lambda }_1)i}\right]}{\ln n}.$$

$$\alpha (\mathcal{T},{\lambda }_1,\lambda )=\beta (\mathcal{T},{\lambda }_1,\lambda )=\frac{{\lambda }^2+(b-1)\lambda {\lambda }_1+{\lambda }_1}{1+b{\lambda }_1}$$

$$V^u:=\left\{v\in V(T):u\le v\right\}.$$

$$br(\mathcal{T})=\sup \left\{\gamma >0:\underset{\pi \in \Pi }{\inf }\sum _{e\in \pi }{\gamma }^{-|e|}>0\right\}\in [1,\infty ].$$

$$b{r}_r(\mathcal{T})=\sup \left\{\gamma >0:\underset{\pi \in \Pi }{\inf }\sum _{e\in \pi }|e{|}^{-\gamma }>0\right\}\in [0,\infty ].$$

$$\underline{gr(\mathcal{T})}=\mathrm{liminf}{\left|{\mathcal{T}}_n\right|}^{\frac{1}{n}}.$$

$$\overline{gr(\mathcal{T})}=\mathrm{limsup}{\left|{\mathcal{T}}_n\right|}^{\frac{1}{n}}.$$

$$\mathbb{P}(X_0=\varrho )=1,$$

$$\mathbb{P}\left(X_{n+1}=(X_n{)}_i|X_0,\cdots ,X_n\right)=\{\begin{array}{c}\frac{{\lambda }_j}{1+\partial (X_n){\lambda }_j}\text{ if }j\le M\\ \frac{\lambda }{1+\partial (X_n)\lambda }\text{ if }j>M\end{array}$$

$$\mathbb{P}\left(X_{n+1}=X_n^{-1}|X_0,\cdots ,X_n\right)=\{\begin{array}{c}\frac{1}{1+\partial (X_n){\lambda }_j}\text{ if }j\le M\\ \frac{1}{1+\partial (X_n)\lambda }\text{ if }j>M\end{array}$$

$$T(\nu ):=\inf \{n\ge 1:X_n=\nu \}.$$

$$\mathbb{P}\left(T(\varrho )=\infty \right)>0.$$

$$\begin{array}{rl}\psi (e,\lambda )& =\frac{{\lambda }^{|e|-1}-1}{{\lambda }^{|e|}-1}\text{ if }\lambda \ne 1,\\ \psi (e,\lambda )& =\frac{|e|-1}{|e|}\text{ if }\lambda =1.\end{array}$$

$$\varphi (e,{\lambda }_1,\lambda )=\frac{{\lambda }_1}{1+\partial (e^{-}){\lambda }_1}+\frac{1}{1+\partial (e^{-}){\lambda }_1}\psi (e,\lambda )\psi (e^{-1},\lambda )+\frac{(\partial (e^{-})-1){\lambda }_1}{1+\partial (e^{-}){\lambda }_1}\psi (e,\lambda )$$

$$\Psi (e,{\lambda }_1,\lambda )=\prod _{g\le e}\varphi (g,{\lambda }_1,\lambda ).$$

$$\exists M\in \mathbb{N}\text{ such that }\sup \{\mathrm{deg}\nu :\nu \in V\}\le M.$$

$$RT(\mathcal{T},\mathbf{X})=\sup \{\gamma >0:\underset{\pi \in \Pi }{\inf }\sum _{e\in \pi }{\left(\Psi (e)\right)}^{\gamma }>0\}.$$

$$br(\mathcal{T})=br(\tilde{\mathcal{T}})=2.$$

$$b{r}_r(\mathcal{T})=b{r}_r(\tilde{\mathcal{T}})=\infty .$$

$$\alpha \left(\tilde{\mathcal{T}},1,\frac{\sqrt{3}-1}{2}\right)=\beta \left(\tilde{\mathcal{T}},1,\frac{\sqrt{3}-1}{2}\right)>\frac{1}{2}.$$

$$\mathbb{P}(T(\varrho )<\infty )\le \mathbb{P}(S(\varrho )<\infty ).$$

$$\Psi (e,{\lambda }_1,\lambda )=\left(\prod _{g\le e,|g|>1}\frac{{\lambda }^2+(\partial (g^{-})-1){\lambda }_1\lambda +{\lambda }_1}{1+\partial (g^{-}){\lambda }_1}\right)\prod _{g\le e,|g|>1}\left(\frac{1-{\lambda }^{|g|}\left(\frac{1+\partial (g^{-}){\lambda }_1}{{\lambda }^2+(\partial (g^{-})-1){\lambda }_1\lambda +{\lambda }_1}\right)}{1-{\lambda }^{|g|}}\right).$$

$$\Psi (e,{\lambda }_1,\lambda )=\prod _{g\le e,|g|>1}\left(1-\frac{(\partial (g^{-})-1){\lambda }_1+2}{|g|\left(1+\partial (g^{-}){\lambda }_1\right)}\right).$$

$$\Psi (e,{\lambda }_1,\lambda )=\prod _{g\le e,|g|>1}\varphi (g,{\lambda }_1,\lambda )$$

$$=\prod _{g\le e,|g|>1}\left(\frac{{\lambda }_1}{1+\partial (g^{-}){\lambda }_1}+\frac{1}{1+\partial (g^{-}){\lambda }_1}\psi (e,\lambda )\psi (e^{-1},\lambda )+\frac{(\partial (g^{-})-1){\lambda }_1}{1+\partial (g^{-}){\lambda }_1}\psi (e,\lambda )\right)$$

$$=\left(\prod _{g\le e,|g|>1}\frac{1}{1+\partial (g^{-}){\lambda }_1}\right)\prod _{g\le e,|g|>1}\left({\lambda }_1+\psi (e,\lambda )\psi (e^{-1},\lambda )+(\partial (g^{-})-1){\lambda }_1\psi (e,\lambda )\right)$$

$$\begin{array}{rl}& {\lambda }_1+\psi (g,\lambda )\psi (g^{-1},\lambda )+(\partial (g^{-})-1){\lambda }_1\psi (g,\lambda )\\ =& {\lambda }_1+\left(\frac{1-(1/\lambda {)}^{|g|-2}}{1-(1/\lambda {)}^{|g|}}\right)+\left((\partial (g^{-})-1){\lambda }_1\frac{1-(1/\lambda {)}^{|g|-1}}{1-(1/\lambda {)}^{|g|}}\right)\\ =& {\lambda }_1+\left(\frac{{\lambda }^{|g|}-{\lambda }^2}{{\lambda }^{|g|}-1}\right)+(\partial (g^{-})-1){\lambda }_1\left(\frac{{\lambda }^{|g|}-\lambda }{{\lambda }^{|g|}-1}\right)\\ =& \frac{{\lambda }^2+(\partial (g^{-})-1){\lambda }_1\lambda +{\lambda }_1-{\lambda }^{|g|}\left(1+\partial (g^{-}){\lambda }_1\right)}{1-{\lambda }^{|g|}}\\ =& \left({\lambda }^2+(\partial (g^{-})-1){\lambda }_1\lambda +{\lambda }_1\right)\left(\frac{1-{\lambda }^{|g|}\left(\frac{1+\partial (g^{-}){\lambda }_1}{{\lambda }^2+(\partial (g^{-})-1){\lambda }_1\lambda +{\lambda }_1}\right)}{1-{\lambda }^{|g|}}\right).\end{array}$$

$$\Psi (e,{\lambda }_1,\lambda )=\prod _{g\le e,|g|>1}\varphi (g,{\lambda }_1,\lambda )$$

Full text

We keep using the techniques introduced by Collevecchio, Kious and Sidoravicius [5] with the aim of investigating the phase transition of Once-excited random walk on general trees.

In addition, we prove if $\mathcal{T}$ is a tree whose branching number is larger than $1$, any multi-excited random walk on $\mathcal{T}$ moving, after excitation, like a simple random walk is transient.

1. Introduction

In this paper, we study a particular case of multi-excited random walks on trees, introduced by Volkov [11], called the once-excited random walk.

Let $M\in\mathbb{N}$, $(\lambda_{1},...,\lambda_{M})\in(\mathbb{R}_{+})^{M}$ and $\lambda>0$. Let $\mathcal{T}$ be an infinite, locally-finite, tree rooted at $\varrho$. The $(\lambda_{1},...,\lambda_{M},\lambda)$-ERW on $\mathcal{T}$, is a nearest-neighbor random walk $(X_{n})$ started at $\varrho$ such that if $X_{n}$ is on a site for the $i$-th time for $i\leq M$, then the walker takes a random step of a biased random walk with bias $\lambda_{i}$ (i.e. it jumps on its parent with probability proportional to $1$, or jumps on a particular offspring of $\nu$ with probability proportional to $\lambda_{i}$); and if $i>M$, then $X_{n}$ takes a random step of a biased random walk with bias $\lambda$. In the case $M=1$, it is called the once-excited random walk with parameters $(\lambda_{1},\lambda)$. We write $(\lambda_{1},\lambda)$-OERW for $(\lambda_{1},\lambda)$-ERW. The definition of the model and the vocabulary will be made clear in Section 2.3.

Unlike the case of once-reinforced random walk in [5] or digging-random walk in [4], the phase transition of OERW does not depend only on the branching-ruin number and the branching number of tree (see Section 4 for more details). In the case $\mathcal{T}$ is a spherically symmetric tree, we give a sharp phase transition recurrence/transience in terms of their branching number and branching-ruin number and others.

In the following, we denote $br(\mathcal{T})$ the branching number of a tree $\mathcal{T}$ and $br_{r}(\mathcal{T})$ the branching-ruin number of a tree $\mathcal{T}$, see (2.1) and (2.2) for their definitions. Let us simply emphasize that, for any tree $\mathcal{T}$, its branching number is at least one, i.e. $br(\mathcal{T})\geq 1$, whereas the branching-ruin number is nonnegative, i.e. $br_{r}(\mathcal{T})\geq 0$.

A tree $\mathcal{T}$ is said to be spherically symmetric if for every vertex $\nu$, $\deg\nu$ depends only on $\left|\nu\right|$, where $\left|\nu\right|$ denote its distance from the root and $\deg\nu$ is its number of neighbors. Let $\mathcal{T}$ be a spherically symmetric tree. For any $n\geq 0$, let $x_{n}$ be the number of children of a vertex at level $n$. For any $\lambda_{1}\geq 0$ and $\lambda>0$, we define the following quantities:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Theorem 1**.**

Let $\mathcal{T}$ be a spherically symmetric tree, and let $\lambda_{1}\geq 0$, $\lambda>0$. Denote ${\bf X}$ the $(\lambda_{1},\lambda)$-OERW on $\mathcal{T}$. Assume that there exists a constant $M>0$ such that $\sup_{\nu\in V}\deg\nu\leq M$, then we have

1. (1)

in the case $\lambda=1$, if $\eta(\mathcal{T},\lambda_{1})<br_{r}(\mathcal{T})$ then ${\bf X}$ is transient and if $\gamma(\mathcal{T},\lambda_{1})>br_{r}(\mathcal{T})$ then ${\bf X}$ is recurrent; 2. (2)

assume that $\lambda_{1}\geq 0$, $\lambda\neq 1$ and $br(\mathcal{T})>1$, if $\beta(\mathcal{T},\lambda_{1},\lambda)<\frac{1}{br(\mathcal{T})}$ then ${\bf X}$ is recurrent and if $\alpha(\mathcal{T},\lambda_{1},\lambda)>\frac{1}{br(\mathcal{T})}$ then ${\bf X}$ is transient.

Note that, for a $b$-ary tree, we have $br(\mathcal{T})=b$ and

[TABLE]

and our result therefore agrees with Corollary 1.6 of [1]. In [1], the authors prove that the walk is recurrent at criticality on regular trees, but this is not expected to be true on any tree). For instance, if $\lambda_{1}=\lambda$, the $(\lambda,\lambda)$-OERW $\bf X$ is the biased random walk with parameter $\lambda$. Therefore $\bf X$ may be recurrent or transient at criticality (see [2], proposition 22).

Volkov [11] conjectured that, any cookie random walk which moves, after excitation, like a simple random walk (i.e. $\lambda=1$) is transient on any tree containing the binary tree. This conjecture was proved by Basdevant and Singh [1]. Here, we extend this conjecture to any tree $\mathcal{T}$ whose branching number is larger than $1$:

Theorem 2**.**

Let $(\lambda_{1},...,\lambda_{M})\in(\mathbb{R}_{+})^{M}$ and consider $(\lambda_{1},...,\lambda_{M},1)$-ERW $\bf X$ on an infinite, locally finite, rooted tree $\mathcal{T}$. If $br(\mathcal{T})>1$, then $\bf X$ is transient.

The techniques used our paper rely on the strategy adopted in [5] or [4]. In particular, for the proof of transience, we here too view the set of edges crossed by ${\bf X}$ before returning to $\varrho$ as the cluster of the root in a particular correlated percolation.

There are two key ingredients that allow us to use the rest of the strategy from [5]. First, we need to define extensions of ${\bf X}$, which are a family of coupled continuous-time versions of ${\bf X}$ defined on subtrees of $\mathcal{T}$. As in [5], we do this through Rubin’s construction in Section 7. But we will see in Section 7, this construction is actually very different to a once-reinforced random walk in [5] or $M$-digging random walk in [4].

Second, we need to prove that the correlated percolation mentioned above is in fact a quasi-independent percolation, see Lemma 17. From there, the problem boils down to proving that a certain quasi-independent percolation is supercritical.

We refer to Theorem 4 for the more general result on a general tree.

2. The model

First, we review some basic definitions of graph theory and then we define the model of multi-excited random walk on trees which was introduced by Volkov[11] and then made general by Basdevant and Singh[1].

2.1. Notation

Let $\mathcal{T}=(V,E)$ be an infinite, locally finite, rooted tree with the root $\varrho$.

Given two vertices $\nu,\mu$ of $\mathcal{T}$, we say that $\nu$ and $\mu$ are neighbors, denoted $\nu\sim\mu$, if $\{\nu,\mu\}$ is an edge of $\mathcal{T}$.

Let $\nu,\mu\in V\setminus\{\varrho\}$, the distance between $\nu$ and $\mu$, denoted by $d(\nu,\mu)$, is the minimum number of edges of the unique self-avoiding paths joining $x$ and $y$. The distance between $\nu$ and $\varrho$ is called height of $\nu$, denoted by $|\nu|$. The parent of $\nu$ is the vertex $\nu^{-1}$ such that $\nu^{-1}\sim\nu$ and $|\nu^{-1}|=|\nu|-1$. We also call $\nu$ is a child of $\nu^{-1}$.

For any $\nu\in V$, denote by $\partial(\nu)$ the number of children of $\nu$ and $\{\nu_{1},...,\nu_{\partial\nu}\}$ is the set of children of $\nu$. We define an order on $\mathcal{T}$ by the following way. For all $\nu$ and $\mu$, we say that $\nu\leq\mu$ if the unique self-avoiding path joining $\varrho$ and $\mu$ contains $\nu$, and we say that $\nu<\mu$ if moreover $\nu\neq\mu$.

Denote by $\mathcal{T}_{n}$ the set of vertices of $\mathcal{T}$ at height $n$. For any $\nu\in\mathcal{T}$, denote by $\mathcal{T}^{\nu}$ the biggest sub-tree of $\mathcal{T}$ rooted at $\nu$, i.e. $\mathcal{T}^{u}=\mathcal{T}[V^{u}]$, where

[TABLE]

For any edge $e$ of $\mathcal{T}$, denote by $e^{+}$ and $e^{-}$ its endpoints with $|e^{+}|=|e^{-}|+1$, and we define the height of $e$ as $|e|=|e^{+}|$.

For two edges $e$ and $g$ of $\mathcal{T}$, we write $g\leq e$ if $g^{+}\leq e^{+}$ and $g<e$ if moreover $g^{+}\neq e^{+}$. For two vertices $\nu$ and $\mu$ of $\mathcal{T}$ such that $\nu<\mu$, we denote by $[\nu,\mu]$ the unique self-avoiding path joining $\nu$ to $\mu$. For two neighboring vertices $\nu$ and $\mu$, we use the slight abuse of notation $[\nu,\mu]$ to denote the edge with endpoints $\nu$ and $\mu$ (note that we allow $\mu<\nu$).

For two edges $e_{1}$ and $e_{2}$ of $E$, denote by $e_{1}\wedge e_{2}$ the vertex with maximal distance from the root such that $e_{1}\wedge e_{2}\leq e_{1}^{+}$ and $e_{1}\wedge e_{2}\leq e_{2}^{+}$.

Finally, we define a particular class of trees, which is called superperiodic tree. Let $\mathcal{T}_{1}=(V_{1},E_{1})$ and $\mathcal{T}_{2}=(V_{2},E_{2})$ be two trees. A morphism of $\mathcal{T}_{1}$ to $\mathcal{T}_{2}$ is a map $f:\mathcal{T}_{1}\rightarrow\mathcal{T}_{2}$ such that whenever $\nu$ and $\mu$ and $\mu$ are incident in $\mathcal{T}_{1}$, then so are $f(\nu)$ and $f(\mu)$ in $\mathcal{T}_{2}$.

Let $N\geq 0$. An infinite, locally finite and rooted tree $\mathcal{T}$ with the root $\varrho$, is said to be $N$-superperiodic if for every $\nu\in V(\mathcal{T})$, there exists an injective morphism $f:\mathcal{T}\rightarrow\mathcal{T}^{f(o)}$ with $f(o)\in\mathcal{T}^{\nu}$ and $|f(o)|-|\nu|\leq N$. A tree $\mathcal{T}$ is called superperiodic tree if there exists $N\geq 0$ such that it is $N$-superperiodic.

2.2. Some quantities on trees

In this section, we review the definitions of branching number, growth rate and branching-ruin number. We refer the reader to ([6] , [8]) for more details on the branching number and growth rate and [5] for more details on the branching-ruin number.

In order to define the branching number and the branching-ruin number of a tree, we will need the notion of cutsets.

Let $\mathcal{T}$ be an infinite, locally finite and rooted tree. A cutset in $\mathcal{T}$ is a set $\pi$ of edges such that every infinite simple path from $a$ must include an edge in $\pi$. The set of cutsets is denoted by $\Pi$.

The branching number of $\mathcal{T}$ is defined as

[TABLE]

The branching-ruin number of $\mathcal{T}$ is defined as

[TABLE]

These quantities depend on the structure of the tree. If $\mathcal{T}$ is spherically symmetric, then there is really no information in the tree than that contained in the sequence $\left(|\mathcal{T}_{n}|,n\geq 0\right)$. Therefore, a tree which is spherically symmetric and whose $n$ generation grows like $b^{n}$ (resp. $n^{b}$), for $b\geq 1$, has a branching number (resp. branching-ruin number) equal to $b$. For more general trees, this becomes more complicated. In the other word, there exists a tree whose $n$ generation grows like $b^{n}$ (resp. $n^{b}$), for $b\geq 1$, but its branching number (resp. branching-ruin number) is not equal to $b$. For instance, the tree 1-3 in ([8], page 4) is an example.

Finally, we review the definition of growth rate of an infinite, locally finite and rooted tree $\mathcal{T}$. Define the lower growth rate of $\mathcal{T}$ by

[TABLE]

Similarly, we can define upper growth rate of $\mathcal{T}$ by

[TABLE]

In the case $\overline{gr(\mathcal{T})}=\underline{gr(\mathcal{T})}$, we define the growth rate of $\mathcal{T}$, denoted by $gr(\mathcal{T})$, by taking the common value of $\overline{gr(\mathcal{T})}$ and $\underline{gr(\mathcal{T})}.$

Now, we state a relationship between the branching number and growth rate of a superperiodic tree.

Theorem 3** (see [8]).**

Let $\mathcal{T}$ be a $N$-superperiodic tree with $\overline{gr}(\mathcal{T})<\infty$. Then the growth rate of $\mathcal{T}$ ​​exists and $gr(\mathcal{T})=br(\mathcal{T})$. Moreover, we have $|\mathcal{T}_{n}|\leq gr(\mathcal{T})^{n+N}$.

2.3. Definition of the model

Now, we define the model of multi-excited random walk on trees. Let $\mathcal{C}=(\lambda_{1},...,\lambda_{M};\lambda)\in\left(\mathbb{R}_{+}\right)^{M}\times\mathbb{R}_{+}^{*}$ and $\mathcal{T}=(V,E)$ be an infinite, locally finite and rooted tree with the root $\varrho$. A $\mathcal{C}$ multi-excited random walk is a stochastic process $\bf X$ $:=(X_{n})_{n\geq 0}$ defined on some probability space, taking the values in $\mathcal{T}$ with the transition probability defined by:

[TABLE]

[TABLE]

[TABLE]

where $i\in\{1,\cdots,k\}$ and $j=|\{0\leq k\leq n:X_{k}=X_{n}\}|$.

We have some particular cases:

• •

If $\mathcal{C}=(0,...,0;\lambda)$, then $\mathcal{C}$ multi-excited random walk is $M$-digging random walk with parameter $\lambda$ ($M$-DRWλ), which was studied in [4].

• •

If $M=0$, then $\mathcal{C}$ multi-excited random walk is the biased random walk with parameter $\lambda$, which was studied in [7].

• •

If $\mathcal{C}=(\lambda_{1};\lambda)$, then $\mathcal{C}$ multi-excited random walk is $(\lambda_{1},\lambda)$-OERW.

The return time of $\bf X$ to a vertex $\nu$ is defined by:

[TABLE]

We say that $\bf X$ is transient if

[TABLE]

Otherwise, we say that $\bf X$ is recurrent.

3. Main results

3.1. Main results about Once-excited random walk

Let $\lambda_{1}\geq 0$ and $\lambda>0$ and we consider the model $(\lambda_{1},\lambda)$-OERW on an infinite, locally finite and rooted tree $\mathcal{T}$. First, we define the following functions. For any $e\in E$, we set $\psi(e,\lambda)=1$ and $\phi(e,\lambda_{1},\lambda)=1$ if $|e|=1$ and, for any $e\in E$ with $|e|>1$, we set

[TABLE]

[TABLE]

Finally, for any $e\in E$, we define:

[TABLE]

We refer the reader to Lemma 14 for the probabilistic interpretation of these functions.

In the following, we assume that

[TABLE]

Let us define the quantity $RT(\mathcal{T},{\bf X})$ which was introduced in [5]:

[TABLE]

Theorem 4**.**

Consider an $(\lambda_{1},\lambda)$-OERW on an infinite, locally finite, rooted tree $\mathcal{T}$, with parameters $\lambda_{1}\geq 0$ and $\lambda>0$. If $RT(\mathcal{T},{\bf X})<1$ then ${\bf X}$ is recurrent. If $RT(\mathcal{T},{\bf X})>1$ and if (3.4) holds, then ${\bf X}$ is transient.

In the following, we consider the case $\mathcal{T}$ is spherically symmetric.

Lemma 5**.**

Consider a $(\lambda_{1},\lambda)$-OERW ${\bf X}$ on a spherically symmetric $\mathcal{T}$, with parameters $\lambda_{1}\geq 0$ and $\lambda>0$. Assume that there exists a constant $M>0$ such that $\sup_{\nu\in V}\deg\nu\leq M$. We have that

1. (1)

in the case $\lambda=1$, if $\eta(\mathcal{T},\lambda_{1})<br_{r}(\mathcal{T})$ then $RT(\mathcal{T},{\bf X})>1$ and if $\gamma(\mathcal{T},\lambda_{1})>br_{r}(\mathcal{T})$ then $RT(\mathcal{T},{\bf X})<1$; 2. (2)

assume that $\lambda_{1}\geq 0$, $\lambda\neq 1$ and $br(\mathcal{T})>1$, if $\beta(\mathcal{T},\lambda_{1},\lambda)<\frac{1}{br(\mathcal{T})}$ then $RT(\mathcal{T},{\bf X})<1$ and if $\alpha(\mathcal{T},\lambda_{1},\lambda)>\frac{1}{br(\mathcal{T})}$ then $RT(\mathcal{T},{\bf X})>1$.

Note that Theorem 1 is a consequence of Theorem 4 and Lemma 5.

3.2. Main results about critical $M$-Digging random walk

Let $M\in\mathbb{N}^{*}$, $\lambda>0$ and we consider the model $M$-DRWλ on an infinite, locally finite and rooted tree $\mathcal{T}$. In [4], Collevecchio-Huynh-Kious was proved that there is a phase transition with respect to the parameter $\lambda$, i.e there exists a critical parameter $\lambda_{c}$. A natural question that arises: what happens if $\lambda=\lambda_{c}$? As we said in the introduction, there is no a good answer for this question.

In [1], Basdevant-Singh proved the critical $M$-digging random walk is recurrent on the regular trees. In this paper, we prove the critical $M$-digging random walk is still recurrent on a particular class of trees which contains the regular trees.

Theorem 6**.**

Let $M\in\mathbb{N}^{*}$ and $\mathcal{T}$ be a superperiodic tree whose upper-growth rate is finite. Then the critical $M$-digging random walk on $\mathcal{T}$ is recurrent.

4. An example

In this section, we give an example to prove that the phase transition of once-excited random walk $(\lambda_{1},\lambda)-OERW$ on a tree $\mathcal{T}$ does not depend only on the branching-ruin number and the branching number of $\mathcal{T}$.

If $\mathcal{T}$ is a spherically symmetric tree, recall that $x_{n}(\mathcal{T})$ is the number of children of a vertex at level $n$.

Let $\mathcal{T}$ (resp. $\mathcal{\widetilde{T}}$) be a spherically symmetric such that for any $n\geq 0$, we have $x_{n}(\mathcal{T})=2$ (resp. $x_{n}(\mathcal{\widetilde{T}})=1$ if $n$ is odd and $x_{n}(\mathcal{\widetilde{T}})=4$ if not). Then we obtain :

[TABLE]

Lemma 7**.**

Consider a $(1,(\sqrt{3}-1)/2)$-OERW  $\bf X$ (resp. $\bf\widetilde{X}$) on $\mathcal{T}$ (resp. $\mathcal{\widetilde{T}}$). Then $\bf X$ is recurrent, but $\bf\widetilde{X}$ is transient.

Proof.

Note that $\mathcal{T}$ is a binary tree, then we can apply Corollary 1.6 of [1] to imply that $\bf X$ is recurrent. On the other hand, by a simple computation we have

[TABLE]

By Theorem 1 and 4.3, we obtain $\bf\widetilde{X}$ is transient. ∎

5. Proof of Theorem 2

Lemma 8**.**

Let $\mathcal{T}$ be an infinite, locally finite and rooted tree. If $br(\mathcal{T})>1$ then $br_{r}(\mathcal{T})=+\infty$.

Proof.

See ([4], proof of Lemma 8, Case V). ∎

Lemma 9**.**

Let $(\lambda_{1},...,\lambda_{M})\in\left(\mathbb{R}_{+}\right)^{M}$ and $\mathcal{T}$ be an infinite, locally finite and rooted tree. If $M$-DRW1 is transient, then $(\lambda_{1},...,\lambda_{M},1)$-ERW is transient.

Proof.

See ([1], Section 3). ∎

Remark 10**.**

Let $T(\varrho)$ (resp. $S(\varrho)$) the return of of $M$-DRW1 (resp. $(\lambda_{1},...,\lambda_{M},1)$-ERW) to the root $\varrho$ of $\mathcal{T}$. It is simple to see that

[TABLE]

Proposition 11**.**

Let $(\lambda_{1},...,\lambda_{M})\in(\mathbb{R}_{+})^{M}$ and consider $(\lambda_{1},...,\lambda_{M},1)$-ERW $\bf X$ on an infinite, locally finite, rooted tree $\mathcal{T}$. If $br(\mathcal{T})>1$, then $\bf X$ is transient.

Proof.

Note that if $\lambda_{i}=0$ for all $1\leq i\leq M$ and $\lambda=1$, then $\bf X$ is a $M$-digging random walk with parameter $1$ ($M$-DRW1). On the other hand, we have $(\lambda_{1},...,\lambda_{M},1)$-ERW is more transient than $M$-DRW1, i.e if $M$-DRW1 is transient then $(\lambda_{1},...,\lambda_{M},1)$-ERW is transient. We complete the proof by using Lemma (8) and Theorem 2 in [4]. ∎

6. Proof of Lemma 5 and Theorem 1

In this section, we prove Lemma 5. Theorem 1 then trivially follows from Theorem 4.

Lemma 12**.**

Recall the definition of $\Psi(e,\lambda_{1},\lambda)$ as in 3.3. We have that, if $\lambda\neq 1$, for any $|e|>1$,

[TABLE]

and if $\lambda=1$, for any $|e|>1$,

[TABLE]

Proof.

We compute the quantity $\Psi(e,\lambda,\lambda_{1})$ by using (3.1), 3.2 and (3.3). We will proceed by distinguishing two cases.

**Case I: $\lambda\neq 1$.

**By (3.1), 3.2 and (3.3), we have

[TABLE]

[TABLE]

[TABLE]

By 3.1, we have:

[TABLE]

Therefore we obtain 6.1.

**Case II: $\lambda=1$.

**By (3.1), 3.2 and (3.3), we have

[TABLE]

[TABLE]

[TABLE]

By 3.1, we have:

[TABLE]

Therefore we obtain 6.2. ∎

Proof of Lemma 5.

We will proceed by distinguishing a few cases.

**Case I: $\lambda\neq 1$, $br(\mathcal{T})>1$ and $\beta(\mathcal{T},\lambda_{1},\lambda)<\frac{1}{br(\mathcal{T})}$.

**By (2.1), there exists $\delta\in(0,1)$ such that

[TABLE]

As $\beta<\beta^{(1-\delta)}$, there exists $c>0$, for any $n>0$,

[TABLE]

By  6.1 and 6.6, there exists $C>0$ such that for any $\pi\in\Pi$,

[TABLE]

Therefore, by (6.5),

[TABLE]

which implies that $RT(\mathcal{T},{\bf X})<1$.

**Case II: $\lambda\neq 1$, $br(\mathcal{T})>1$ and $\alpha(\mathcal{T},\lambda_{1},\lambda)>\frac{1}{br(\mathcal{T})}$.

**First, note that if $\lambda>1$ and $br(\mathcal{T})>1$ then $\bf X$ is transient. Now, assume that $\lambda<1$, $br(\mathcal{T})>1$ and $\alpha(\mathcal{T},\lambda_{1},\lambda)>\frac{1}{br(\mathcal{T})}$. We have that there exists $\delta>0$ and $\varepsilon>0$ such that

[TABLE]

By 1.1 and $\lambda<1$, we obtain $\alpha<1$, therefore $\alpha^{1+\delta}<\alpha$. We have that there exists $c>0$, for any $n>0$,

[TABLE]

By  6.1 and 6.10, there exists $C>0$ such that for any $\pi\in\Pi$,

[TABLE]

Therefore, by (6.9),

[TABLE]

which implies that $RT(\mathcal{T},{\bf X})>1$.

**Case III: $\lambda=1$ and $\eta(\mathcal{T},\lambda_{1})<br_{r}(\mathcal{T})$.

**We have that there exists $\delta>0$ and $\varepsilon>0$ such that

[TABLE]

As $\eta<(1+\delta)\eta$, by 1.4 there exists $c>0$, for any $n>0$,

[TABLE]

By  6.2 and 6.14, there exists $C>0$ such that for any $\pi\in\Pi$,

[TABLE]

Therefore, by (6.13),

[TABLE]

which implies that $RT(\mathcal{T},{\bf X})>1$.

**Case IV: $\lambda=1$ and $\gamma(\mathcal{T},\lambda_{1})>br_{r}(\mathcal{T})$

**We have that there exists $\delta>0$ such that

[TABLE]

As $\eta>(1-\delta)\eta$, by 1.4 there exists $c>0$, for any $n>0$,

[TABLE]

By  6.2 and 6.18, there exists $C>0$ such that for any $\pi\in\Pi$,

[TABLE]

Therefore, by (6.17),

[TABLE]

which implies that $RT(\mathcal{T},{\bf X})<1$. ∎

7. Extensions

First of all, let us describe the dynamic of this model. If $\bf X$ visits a vertex $\nu$ for the first time, three cases can occur for visiting $\nu_{1}$ (see Figure 1):

• •

It eats the cookie at $\nu$ and returns to the parent of $\nu$ (i.e. $\nu^{-1}$) with probability $\frac{1}{1+\partial(\nu)\lambda_{1}}$. It then visits $\nu$ for the second time, and goes to $\nu_{1}$ with probability $\frac{\lambda}{1+\partial(\nu)\lambda}$.

• •

It goes directly to $\nu_{1}$ with probability $\frac{\lambda_{1}}{1+\partial(\nu)\lambda_{1}}$.

• •

It goes to one of the chidren of $\nu$ except for $\nu_{1}$, with probability $\frac{(\partial\nu-1)\lambda_{1}}{1+\partial(\nu)\lambda_{1}}$. It then visits $\nu$ for the second time, and goes to $\nu_{1}$ with probability $\frac{\lambda}{1+\partial(\nu)\lambda}$.

Now, we introduce a construction of once-excited random walk by using the Rubin’s construction. Let $(\Omega,\mathcal{F},\mathbf{P})$ denote a probability space on which

[TABLE]

are two families of independent mean $1$ exponential random variables, where $(\nu,\mu)$ denotes an ordered pair of vertices. Let

[TABLE]

is a family of independent uniformly random variables on $[0,1]$ which is independent to $\bf Y$ and $\bf Z$.

For any pair vertices $\nu,\mu\in V$ with $\nu\sim\mu$, we define the following quantities

[TABLE]

Let $\mathcal{T}^{\prime}$ be a sub-tree of $\mathcal{T}$, we define the extension ${\bf X}^{(\mathcal{T}^{\prime})}=(V^{\prime},E^{\prime})$ on $\mathcal{T}^{\prime}$ in the following way. Denote by $\varrho^{\prime}$ the root of $\mathcal{T}^{\prime}$ which be defined as the vertex of $V^{\prime}$ with smallest distance to the root of $\mathcal{T}$. For any family of nonnegative integers $\bar{k}=(k_{\mu})_{\mu:[\nu,\mu]\in E^{\prime}}$, we let

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Set ${X}^{(\mathcal{T}^{\prime})}_{0}=\varrho^{\prime}$ and on the event $A^{(\mathcal{T}^{\prime})}_{\bar{k},n,\nu}\cap\{t_{\nu}(n)\leq 1\}$:

• •

If $U_{\nu}<\frac{1}{1+\partial(\nu)\lambda_{1}}$, then we set ${X}^{(\mathcal{T}^{\prime})}_{n+1}=\nu^{-1}$.

• •

If   $U_{\nu}\in\left[\frac{1+(j-1)\lambda_{1}}{1+\partial(\nu)\lambda_{1}},\frac{1+j\lambda_{1}}{1+\partial(\nu)\lambda_{1}}\right]$ and $j\in\mathcal{I}^{\mathcal{T}}(\nu)$, then we set $X^{(\mathcal{T^{\prime}})}_{n+1}=v_{j}$.

• •

If   $U_{\nu}\in\left[\frac{1+(j-1)\lambda_{1}}{1+\partial(\nu)\lambda_{1}},\frac{1+j\lambda_{1}}{1+\partial(\nu)\lambda_{1}}\right]$ for some $j\notin\mathcal{I}^{\mathcal{T}}(\nu)$ and

[TABLE]

we set $X^{(\mathcal{T^{\prime}})}_{n+1}=\nu^{\prime}$.

On the event

[TABLE]

we set ${X}^{(\mathcal{T}^{\prime})}_{n+1}=\nu^{\prime}$, where the function $r$ is defined in (7.4) and the clocks $Y$’s are from the same collection ${\bf Y}$ fixed in (7.1).

Thus, this defines ${\bf X}^{(\mathcal{T})}$ as the extension on the whole tree. By using the properties of independent exponential random variables, it is easy to check that this construction is a construction of $(\lambda_{1},\lambda)$-OERW on the tree $\mathcal{T}$. We refer the reader to ([4], section 7) for more discussions on this construction.

In the case $\mathcal{T}^{\prime}=[\varrho,\nu]$ for some vertex $\nu$ of $\mathcal{T}$, we write ${\bf X}^{(\nu)}$ instead of ${\bf X}^{([\varrho,\nu])}$, and we denote $T^{(\nu)}(\cdot)$ the return times associated to ${\bf X}^{(\nu)}$. For simplicity, we will also write ${\bf X}^{(e)}$ and $T^{(e)}(\cdot)$ instead of ${\bf X}^{(e^{+})}$ and $T^{(e^{+})}(\cdot)$ for $e\in E$.

Remark 13**.**

Let $\mathcal{T}^{\prime}$ be a proper subtree of $\mathcal{T}$. Note that ${\bf X}^{(\mathcal{T}^{\prime})}$ is not $(\lambda_{1},\lambda)$-OERW on $\mathcal{T}^{\prime}$, that is different with $M$-digging random walk (see [4], section 7) and once-reinforced random walk (see [5], section 5).

Finally, we give a probabilistic interpretation of the functions $\phi$ and $\Psi$:

Lemma 14**.**

For any $e\in E$ and any $g\leq e$, we have

[TABLE]

where $\theta$ is the canonical shift on the trajectories.

Proof.

Let $e\in E$ and $g\leq e$. For simplicity, we set

[TABLE]

[TABLE]

[TABLE]

where $j\in\{1,...,\partial(g^{-})\}$ such that $(g^{-})_{j}=g^{+}$. We have that

[TABLE]

On the other hand, we have the following equalities:

[TABLE]

[TABLE]

[TABLE]

We use (7.13), (7.14), (7.15) and (7.16) to obtain the results. ∎

8. Recurrence in Theorem 4: The case $RT(\mathcal{T},{\bf X})<1$

Proposition 15**.**

If

[TABLE]

then $\mathbf{X}$ is recurrent.

Proof.

The proof is identical to the proof of Proposition 10 of [5]. ∎

9. Transience in Theorem 4: The case $RT(\mathcal{T},{\bf X})>1$

In order to prove transience, we use the relationship between the walk $\bf{X}$ and its associated percolation.

9.1. Link with percolation

Denote by $C(\varrho)$ the set of edges which are crossed by ${\bf X}$ before returning to $\varrho$, that is:

[TABLE]

We define an other percolation which will be more easy to study. In order to do this, we use the Rubin’ s construction and the extensions introduced in Section 7. We define

[TABLE]

We say that an edge $e\in E$ is open if and only if $e\in\mathcal{C}_{CP}(\varrho)$.

Lemma 16**.**

We have that

[TABLE]

Proof.

We can follow line by line the proof of Lemma 11 in [5]. ∎

For simplicity, for a vertex $v\in V$, we write $v\in\mathcal{C}_{\mathrm{CP}}(\varrho)$ if one of the edges incident to $v$ is in $\mathcal{C}_{\mathrm{CP}}(\varrho)$. Besides, recall that for two edges $e_{1}$ and $e_{2}$, their common ancestor with highest generation is the vertex denoted $e_{1}\wedge e_{2}$.

Lemma 17**.**

Let $\lambda_{1}\geq 0$, $\lambda>0$ and $\mathcal{T}$ be an infinite, locally finite and rooted tree with the root $\varrho$. Assume that the condition (3.4) holds with some constant $M$. Then the correlated percolation induced by $\mathcal{C}_{CP}$ is quasi-independent, i.e. there exists a constant $C_{Q}\in(0,+\infty)$ such that, for any two edges $e_{1},e_{2}$, we have that

[TABLE]

Proof.

Recall the construction of Section 7. Note that if $e_{1}\wedge e_{2}=\varrho$, then the extensions on $[\varrho,e_{1}]$ and $[\varrho,e_{2}]$ are independent, then the conclusion of Lemma holds with $C=1$. Assume that $e_{1}\wedge e_{2}\neq\varrho$, and note that the extensions on $[\varrho,e_{1}]$ and $[\varrho,e_{2}]$ are dependent since they use the same clocks on $[\varrho,e_{1}\wedge e_{2}]$. Denote by $e$ the unique edge of $\mathcal{T}$ such that $e^{+}=e_{1}\wedge e_{2}$. For $i\in\{1,2\}$, let $v_{i}$ be the vertex which is the offspring of $e^{+}$ lying the path from $\varrho$ to $e_{i}$. Note that $v_{i}$ could be equal to $e^{+}_{i}$. Let $i_{1}$ (resp. $i_{2}$) be an element of $\{1,...,\partial(e^{+})\}$ such that $(e^{+})_{i_{1}}=v_{1}$ (resp. $(e^{+})_{i_{2}}=v_{2}$).

As the events $\{e\in\mathcal{C}_{CP}\}$ and $U_{e_{1}\wedge e_{2}}$ are independent, therefore:

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

[TABLE]

In the same way, for any $j\in\{1,2\}$, we have:

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

Lemma 18**.**

There exists four constants $(\alpha_{1},\alpha_{2},\alpha_{3},\alpha)$ depend on $\mathcal{T}$, $\lambda$ and $\lambda_{1}$ such that:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

We deduce from Lemma 18 that

[TABLE]

where $\alpha=\max_{i\in\{1,2,3,4\}}\alpha_{i}$. The latter inequality concludes the proof of Proposition. ∎

Proof of Lemma 18.

Now, we will adapt the argument from the proof of Lemma 12 in [5]. We prove that there exists $\alpha_{1}$ such that $A\leq\alpha_{1}E_{1}E_{2}$ and we use the same argument for the other inequalities.

First, by using condition 3.4, note that,

[TABLE]

[TABLE]

On the event $\left\{e\in\mathcal{C}_{CP}(\varrho),\,U_{e^{+}}<\frac{1}{1+\partial(e^{+})\lambda_{1}}\right\}$ we have $X^{(e)}_{T^{(e)}(e^{+})+1}=e^{-}$. We then define $\widetilde{T}^{(e)}(e^{+}):=\inf\left\{n\geq T^{(e)}(e^{+})+1:X^{(e)}_{n}=e^{+}\right\}$. We define the following quantities:

[TABLE]

where $|A|$ denotes the cardinality of a set $A$ and $\theta$ is the canonical shift on trajectories. Note that $L(e)$ is the time consumed by the clocks attached to the oriented edge $(e^{+},e^{-})$ before ${\bf X}^{{{\scriptscriptstyle{({e}})}}}$, ${X}^{{{\scriptscriptstyle{({e_{1}}})}}}$ or ${X}^{{{\scriptscriptstyle{({e_{2}}})}}}$ goes back to ${\varrho}$ once it has returned $e^{+}$ after the time $T^{(e)}(e^{+})$. Recall that these three extensions are coupled and thus the time $L(e)$ is the same for the three of them.

For $i\in\{1,2\}$, recall that $v_{i}$ is the vertex which is the offspring of $e^{+}$ lying the path from $\varrho$ to $e_{i}$. Note that $v_{i}$ could be equal to $e^{+}_{i}$. We define for $i\in\{1,2\}$:

[TABLE]

Here, $L^{*}(e_{i})$, $i\in\{1,2\}$, is the time consumed by the clocks attached to the oriented edge $(e^{+},v_{i})$ before ${\bf X}^{{{\scriptscriptstyle{({e_{i}}})}}}$, or ${\bf X}^{[e^{+},e_{i}^{+}]}$, hits $e_{i}^{+}$.

Notice that the three quantities $L(e)$, $L^{*}(e_{1})$ and $L^{*}(e_{2})$ are independent, and we also have:

[TABLE]

[TABLE]

[TABLE]

Now, the random variable $N(e)$ is simply a geometric random variable (counting the number of trials) with success probability $\lambda^{1-|e|}/\sum_{g\leq e}\lambda^{1-|g|}$. The random variable $N(e)$ is independent of the family $Y(e^{+},e^{-},\cdot)$. As $Y(e^{+},e^{-},j)$ are independent exponential random variable for $j\geq 0$, we then have that $L(e)$ is an exponential random variables with parameter

[TABLE]

A priori, $L^{*}(e_{1})$ and $L^{*}(e_{2})$ are not exponential random variable, but they have a continuous distribution. Denote $f_{1}$ and $f_{2}$ respectively the densities of $L^{*}(e_{1})$ and $L^{*}(e_{2})$. Then, we have that

[TABLE]

Thus, one can write

[TABLE]

Note that:

[TABLE]

where $\widetilde{L}(e)$ is an exponential variable with parameter $p/2$. Note that, in view of (9.22), $\widetilde{L}(e)$ has the same law as $L(e)$ when we replace the weight of an edge $g^{\prime}$ by $\lambda^{-|g^{\prime}|+1}/2$ for $g^{\prime}\leq e$ only, and keep the other weights the same.

For simplicity, for any $g\in E$, we set $w(g)=\lambda^{|g|-1}$. For $g\in E$ such that $e<g$, define the functions $\widetilde{\psi}$ and $\widetilde{\phi}$ in a similar way as $\psi$ and $\phi$, except that we replace the weight of an edge $g^{\prime}$ by $\lambda^{-|g^{\prime}|+1}/2$ for $g^{\prime}\leq e$ only, and keep the other weights the same, that is, for $g\in E$, $e<g$,

[TABLE]

We obtain:

[TABLE]

Now, we compute the product:

[TABLE]

[TABLE]

[TABLE]

Lemma 19**.**

There exists a constant $c=c(\lambda_{1},\lambda)$ which do not depend on $e$, $e_{1}$ and $e_{2}$, such that:

[TABLE]

On the other hand, by using Lemma 19, for any $e$ and $e_{1}$ we have that

[TABLE]

By using 9.30, Lemma 19 and condition (3.4), we obtain:

[TABLE]

We have just proved that

[TABLE]

By doing a very similar computation, one can prove that

[TABLE]

Moreover, we have

[TABLE]

The conclusion (9.4) follows by using (9.16), (9.24), (9.34), together with (9.32) and (9.33). ∎

It remains to prove Lemma 19.

Proof of Lemma 19.

By a simple computation, for any $e<g\leq e_{1}$,

[TABLE]

We will proceed by distinguishing three cases.

**Case I: $\lambda<1$.

**By (9.35), we have that

[TABLE]

Hence, there exists a constants $c_{1}$ such that

[TABLE]

Therefore we obtain

[TABLE]

**Case II: $\lambda=1$.

**By (9.35), we have that

[TABLE]

Therefore we obtain

[TABLE]

On the other hand, we have:

[TABLE]

We use (9.40) and (9.41) to obtain the result.

**Case III: $\lambda>1$.

**By (9.35), we have that

[TABLE]

Hence, there exists a constants $c_{2}$ such that

[TABLE]

Therefore we obtain

[TABLE]

∎

9.2. Transience in Theorem 4: The case $RT(\mathcal{T},{\bf X})>1$

Proposition 20**.**

If $RT(\mathcal{T},\mathbf{X})>1$ and if (3.4) is satisfied then ${\bf X}$ is transient.

Proof.

The proof is now easy, we can follow line by line the Appendix A.2 of [4]. ∎

10. Proof of Theorem 6

This section is independent with the previous sections. In this section, we prove a criterion which can apply to the critical $M$-digging random walk on superperiodic trees. We will use the Rubin’s construction (resp. the definition of $\mathcal{C}(\varrho)$, $\mathcal{C}_{CP}(\varrho)$) from section 7 (resp. section 8.1) of [4]. We will allow ourselves to omit these definitions and refer the readers to [4] for more details.

The main idea for the proof of Theorem 6 is that the number of surviving rays of the percolation $\mathcal{C}_{CP}(\varrho)$ almost surely is either zero or infinite. This property was proved in the case of Bernoulli percolation (see [8] proposition 5.27) or target percolation (see [10], lemma 4.2). The main difficulty that we have to face is that the FKG inequality is not true for our percolation.

10.1. Some definitions

Let $\lambda>0$, $M\in\mathbb{N}$ and $\mathcal{T}$ be an infinite, locally finite and rooted tree. For each $v\in V(\mathcal{T})$, recall the definition of subtree $\mathcal{T}^{v}$ of $\mathcal{T}$ from Section 2.1. Let ${\bf X}^{v,\lambda}$ be the $M$-digging random walk on $\mathcal{T}^{v}$. We say that $\mathcal{T}$ is uniformly transient if for any $\lambda$ such that the $M$-digging random walk on $\mathcal{T}$ with parameter $\lambda$ is transient (i.e. ${\bf X}^{\varrho,\lambda}$ is transient),

[TABLE]

It is called weakly uniformly transient if there exists a sequence of finite pairwise disjoint $\pi_{n}$ such that

[TABLE]

where $V(\pi_{n})=\{e^{-}:e\in\pi_{n}\}$.

Remark 21**.**

• •

If $\mathcal{T}$ is uniformly transient, then $\mathcal{T}$ is also weakly uniformly transient, but the reverse is not always true.

• •

The superperiodic trees are uniformly transient.

An infinite self-avoiding path starting at $\varrho$ is called a ray. The set of all rays, denoted by $\partial\mathcal{T}$, is called the boundary of $\mathcal{T}$. Let $\phi:\mathbb{Z}^{+}\rightarrow\mathbb{R}$ be a decreasing positive function with $\phi(n)\rightarrow 0$ as $n\rightarrow\infty$. The Hausdorff mearsure of $\mathcal{T}$ in gauge $\phi$ is

[TABLE]

where the $\liminf$ is taken over $\Pi$ such that the distance from $\varrho$ to the nearest vertex in $\Pi$ goes to infinity. We say that $\mathcal{T}$ has $\sigma$-finite Hausdorff measure in gauge $\phi$ if $\partial\mathcal{T}$ is the union of countably many subsets with finite Hausdorff measure in gauge $\phi$.

Finally, If $\lambda$ is such that the $M$-digging random walk $X$ with parameter $\lambda$ on $\mathcal{T}$ is transient, on the event $\{T(\varrho)=\infty\}$, its path determines an infinite branch in $\mathcal{T}$, which can be seen as a random ray $\omega^{\infty}$, and call it the limit walk of $X$. Equivalently, on the event $\{T(\varrho)=\infty\}$, we define the limit walk as follows: For any $k\geq 1$,

[TABLE]

Note that $\mathbb{P}\left(\omega^{\infty}(0)=\varrho\right)=1$. For any $k\geq 1$, we call the $k$-first steps of $\omega^{\infty}$ is $(\omega^{\infty}(0),\cdots,\omega^{\infty}(k))$, denoted by $\omega^{\infty}_{|{[0,n]}}$.

10.2. Proof of Theorem 6

We begin with the following proposition:

Proposition 22**.**

Let $\bf X$ be a $M$-digging random walk with parameter $\lambda_{c}$ on an uniformly transient tree $\mathcal{T}$ and recall the definition of $\mathcal{C}_{CP}$ from $\bf X$ as in ([4], Section 7). Consider the percolation induced by $\mathcal{C}_{CP}$ and let $\phi(n)=\mathbb{P}(\varrho\leftrightarrow v)$ for $v\in\mathcal{T}_{n}$.

1. (1)

Almost surely, the number of surviving rays is either zero or infinite. 2. (2)

If $\partial\mathcal{T}$ has $\sigma$-finite Hausdorff measure in the gauge $\{\phi(n)\}$, then $\mathbb{P}(\varrho\leftrightarrow\infty)=0$. In particular, $\bf X$ is recurrent.

The overall strategy for the proof of Proposition 22 is as follows. First, if ${\bf X}$ is recurrent, then the percolation induced by $\mathcal{C}_{CP}$ almost surely have no surviving ray. Next, assume that ${\bf X}$ is transient. On the event $\left\{T(\varrho)=\infty\right\}$, the limit walk $\omega^{\infty}$ is a surviving ray of $\mathcal{C}_{CP}(\varrho)$. Given $n\in\mathbb{N}$ and conditioning on $\omega^{\infty}_{|{[0,n]}}$, by using the Rubin’s construction and the definition of uniformly transient, we prove that there exists a surviving ray in $\mathcal{T}^{\omega^{\infty}(i)}$ with probability larger than a constant $c$ which do not depend on $i$ and $\omega^{\infty}$ (see Figure 2). The following basic lemma is necessary:

Lemma 23**.**

Let $\lambda>0$ and $\mathcal{T}$ be an infinite, locally finite and rooted tree. Let $\overline{M}:=(m_{v},v\in V(\mathcal{T}))$ be a family of non-negative integers. Denote by $\bf X$ the $M$-digging random walk with parameter $\lambda$ and $\bf Y$ the $\overline{M}$-digging random walk associated with the inhomogeneous initial number of cookies $\overline{M}$ with parameter $\lambda$ (see [4], section 2.3.2 for more details on the definition of $\overline{M}$-digging random walk). Denote by $T^{\bf X}(\varrho)$ (resp. $T^{\bf Y}(\varrho)$) the return time of $\bf X$ (resp. $\bf Y$) to $\varrho$. Assume that $m(v)\leq M$ for all $v\in V(\mathcal{T})$, we then have

[TABLE]

Proof.

The proof is simple, therefore it is omitted. ∎

Proof of Proposition 22.

Let $\mathcal{A}_{k}$ denote the event that exactly $k$ rays survive and assume that

[TABLE]

Hence,

[TABLE]

By (10.6) and Lemma 22 in [4], we have that:

[TABLE]

and therefore $\bf X$ is transient.

On the event $\{T(\varrho)=\infty\}$, the limit walk $\omega^{\infty}$ of $\bf X$ is well defined and it is a surviving ray. Let $n$ be a positive integer and $\gamma:=(\gamma_{0}=\varrho,\gamma_{1}=v_{1},\cdots,\gamma_{n}=v_{n})$ be a path of length $n$ of $\mathcal{T}$. Denote by $B_{n,\gamma}$ the following event:

[TABLE]

For any $1\leq k\leq n$, define a sub-tree $\mathcal{T}^{v_{i}}$ of $\mathcal{T}$ in the following way (see Figure 2).

• •

The root of $\mathcal{T}^{v_{i}}$ is the vertex $v_{i}$.

• •

If $\partial(v_{i})<2$ then $\mathcal{T}^{v_{i}}$ is a tree with a single vertex $v_{i}$: for example, $\mathcal{T}^{v_{3}}$ in Figure 2.

• •

If $\partial(\gamma_{i})\geq 2$, choose one of its children which is different to $v_{i+1}$, denoted by $v$. We then set:

[TABLE]

Note that for every pair $(i,j)\in[1,n]^{2}$, we have $V(\mathcal{T}^{v_{i}})\cap V(\mathcal{T}^{v_{j}})=\varnothing$.

Now, conditioning on the event $\mathcal{B}_{n,\gamma}$. Let $\ell$ be the last time ${\bf X}$ visits $v_{n}$, i.e.

[TABLE]

By the definition of limit walk, $\ell$ is finite on the event $\mathcal{B}_{n,\gamma}$. For each $i\in[1,n]$ and for all $v\in V(\mathcal{T}^{v_{i}})$, denote by $m^{i}(v)$ the remaining number of cookies at $v$ after time $\ell$, i.e.

[TABLE]

By using the extensions introduced in ([4], Section 7), the next steps on the tree $\mathcal{T}^{v_{i}}$ are given by the digging random walk associated with the inhomogeneous initial number of cookies $(m^{i}(v),v\in V(\mathcal{T}^{v_{i}}))$ and the same parameter $\lambda_{c}$ as $\bf X$, denoted by ${\bf{X}}^{v_{i},m^{i},\lambda_{c}}$ (see [4], section 2.3.2 for more details on the definition of ${\bf{X}}^{v_{i},m^{i},\lambda_{c}}$). Denote by $T^{v_{i},m^{i},\lambda_{c}}$ the return time of ${\bf{X}}^{v_{i},m^{i},\lambda_{c}}$ to the root $v_{i}$ of $\mathcal{T}^{v_{i}}$. By the definition of uniformly transient and Lemma 23, there exists a constant $c>0$ which do not depend on $n$ and $\gamma$ such that for any $i$,

[TABLE]

On the event $\{T^{v_{i},m^{i},\lambda_{c}}<\infty\}$, note that $\mathcal{C}_{CP}$ contains a surviving ray in $\mathcal{T}^{v_{i}}$. By (10.11), we have

[TABLE]

On the other hand, we have $\mathcal{A}_{k}\subset\bigcup_{\gamma:|\gamma|=n}\mathcal{B}_{n,\gamma}$, therefore by (10.12) we obtain:

[TABLE]

Since 10.13 holds for any $n$ then we obtain the following contradiction

[TABLE]

For part (2), the proof is similar to part (ii), Lemma 4.2 in [10]. ∎

In the same method as in the proof of Proposition 22, we can prove the slightly stronger result (the proof of which we omit):

Proposition 24**.**

Let $\bf X$ be a $M$-digging random walk with parameter $\lambda_{c}$ on a weakly uniformly transient tree $\mathcal{T}$ and recall the definition of $\mathcal{C}_{CP}$ from $\bf X$ as in ([4], Section 7). Consider the percolation induced by $\mathcal{C}_{CP}$ and let $\phi(n)=\mathbb{P}(\varrho\leftrightarrow v)$ for $v\in\mathcal{T}_{n}$.

1. (1)

With probability one, the number of surviving rays is either zero or infinite. 2. (2)

If $\partial\mathcal{T}$ has $\sigma$-finite Hausdorff measure in the gauge $\{\phi(n)\}$, then $\mathbb{P}(\varrho\leftrightarrow\infty)=0$. In particular, $\bf X$ is recurrent.

The following corollary is an immediate consequence of Proposition 24.

Corollary 25**.**

Let $M\in\mathbb{N}$ and $\mathcal{T}$ be a weakly uniformly transient tree such that $\partial\mathcal{T}$ has $\sigma$-finite Hausdorff measure in the gauge $\{\phi(n)\}=\left(\frac{1}{br(\mathcal{T})}\right)^{n}$ if $br(\mathcal{T})>1$ and $\{\phi(n)\}=\frac{1}{n^{M+1}}$ if $br(\mathcal{T})=1$. Then the critical $M$-digging random walk on $\mathcal{T}$ is recurrent.

Proposition 26**.**

Let $M\in\mathbb{N}^{*}$ and $\mathcal{T}$ be a superperiodic tree whose upper-growth rate is finite. The critical $M$-digging random walk on $\mathcal{T}$ is recurrent.

Proof.

This is a consequence of Corollary 25 and Theorem 3. ∎

Remark 27**.**

If $M=0$, then $M$-DRWλ is the biased random walk with parameter $\lambda$. The recurrence of critical biased random walk on $\mathcal{T}$ is a consequence of Theorem 3 and Nash-Williams criterion (see [8] or [9]).

Acknowledgement**.**

I am grateful to Daniel Kious and Andrea Collevecchio for precious discussions. I am thankful to Vincent Beffara for his support.