Dimensions of harmonic measures in percolation clusters on hyperbolic groups

Kohki Sakamoto

Abstract.

For the simple random walks in percolation clusters on hyperbolic groups, we show that the associated harmonic measures are exact dimensional and their Hausdorff dimensions are equal to the entropy over the speed. Our method is inspired by cluster relations introduced by Gaboriau and applies to a large class of random environments on the groups.

2020 Mathematics Subject Classification:

32V05

1. Introduction

1.1. Background and main results

A random walk on a word hyperbolic group determines the hitting measure on the Gromov boundary if it has positive speed (cf. [Kai00]). This hitting measure is called the harmonic measure of the random walk. One of the main subjects in the study of such harmonic measures is to establish properties called exact dimensionality and the dimension formula. Let us explain about these terms. In the rest of this section, $\Gamma$ denotes a nonelementary hyperbolic group, and $G$ denotes its Cayley graph with respect to some generating set of $\Gamma$ . For a Borel measure $\nu$ on the boundary $\partial G$ , its upper Hausdorff dimension is defined by

\dim\nu=\underset{\xi\in\partial G}{\nu\text{-}\sup}\liminf_{r\to 0}\frac{\log\nu(B(\xi,r))}{\log r},

where “ $\nu\text{-}\sup$ ” indicates the essential supremum with respect to $\nu$ , and $B(\xi,r)$ denotes the ball centered at $\xi$ with radius $r$ . Informally speaking, it shows the degree of the fractalness of $\nu$ . We say that a harmonic measure $\nu$ is exact dimensional if

\dim\nu=\lim_{r\to 0}\frac{\log\nu(B(\xi,r))}{\log r}

holds for $\nu$ -almost every $\xi$ . Further, we say that $\nu$ satisfies the dimension formula if $\dim\nu$ is equal to $h/l$ , where $h$ and $l$ are the entropy and the speed of the random walk that determines $\nu$ , respectively (see Section 2 for the definition). It has been proved that the harmonic measure of a random walk driven by a single measure on $\Gamma$ is exact dimensional and satisfies the dimension formula, under several types of conditions [Led01], [LP07], [BHM11], [Tan19]. Such results can be used to show that the harmonic measure is singular to another measure on the boundary defined geometrically (see [LP07], [BHM11], and [CLP21]).
On the other hand, there are only a few works concerning harmonic measures associated with random walks in random environments. Lyons, Pemantle, and Peres showed that the harmonic measure of the simple random walk on a supercritical Galton-Watson tree is exact dimensional and satisfies the dimension formula in their influential paper [LPP95]. The notable work that inspires our study is the paper by Kaimanovich [Kai98], where he established the exact dimensionality and the dimension formula for a large class of Markov chains on trees, and it can be applied to random walks in stationary random environments on free groups.
Our main result is a generalization of results in [Kai98]. Namely, we establish exact dimensionality and the dimension formula for random walks in percolation clusters and random conductance models on hyperbolic groups. Here we focus on percolation clusters for simplicity. For $p\in[0,1]$ , the Bernoulli percolation $\mu_{p}$ is defined as the random subgraph of $G$ obtained by independently retaining or deleting each edge with probability $p$ or $1-p$ , respectively. The Bernoulli percolation $\mu_{p}$ is called supercritical if $\mu_{p}$ -almost every $\omega\subset G$ has a connected component with infinite vertices. For supercritical $\mu_{p}$ , we consider the simple random walk on the connected component of $\omega$ containing the identity $1$ (denoted by $C_{\omega}(1)$ ), conditioned on $C_{\omega}(1)$ being infinite. Note that the event of $C_{\omega}(1)$ being infinite, denoted by $\Omega_{1}$ , is $\mu_{p}$ -positive since $\mu_{p}$ is supercritical and $\Gamma$ -invariant. Benjamini, Lyons, and Schramm initiated the study of such random walks in percolation clusters on general Cayley graphs in [BLS99]. They proved that the entropy and the speed (denoted by $h$ and $l$ , respectively) of such random walks are deterministic, i.e., they depend only on $p$ . Further, they showed that $h,l>0$ if the group is nonamenable. Therefore, for supercritical $\mu_{p}$ on $G$ , we can define the harmonic measure associated with the simple random walk on $C_{\omega}(1)$ for $\mu_{p}$ -almost every $\omega$ (recall that every nonelementary hyperbolic group is nonamenable). Based on the ergodic-theoretic approach in [BLS99], we establish the exact dimensionality and the dimension formula for the simple random walks in supercritical Bernoulli percolation clusters.

Theorem 1.1.

Let $\mu_{p}$ be a supercritical Bernoulli percolation on $G$ . Then, for $\mu_{p}$ -almost every $\omega\in\Omega_{1}$ , letting $\nu_{\omega}$ be the harmonic measure on $\partial G$ determined by the simple random walk on $C_{\omega}(1)$ starting from 1, we have

\dim\nu_{\omega}=\lim_{r\to 0}\frac{\log\nu_{\omega}(B(\xi,r))}{\log r}=\frac{h}{l}

for $\nu_{\omega}$ -almost every $\xi\in\partial G$ . In particular, $\dim\nu_{\omega}$ is positive and constant for $\mu_{p}$ -almost every $\omega\in\Omega_{1}$ .

In fact, our strategy works for a more general class of percolation models on $G$ , as follows.

Theorem 1.2.

Let $\mu$ be an ergodic $\Gamma$ -invariant percolation on $G$ having indistinguishable infinite clusters. Assume that the simple random walk on $C_{\omega}(1)$ has positive speed for $\mu$ -almost every $\omega\in\Omega_{1}$ . Then, for $\mu$ -almost every $\omega\in\Omega_{1}$ , letting $\nu_{\omega}$ be the harmonic measure on $\partial G$ determined by the simple random walk on $C_{\omega}(1)$ starting from 1, we have

\dim\nu_{\omega}=\lim_{r\to 0}\frac{\log\nu_{\omega}(B(\xi,r))}{\log r}=\frac{h}{l}

for $\nu_{\omega}$ -almost every $\xi\in\partial G$ . In particular, $\dim\nu_{\omega}$ is positive and constant for $\mu$ -almost every $\omega\in\Omega_{1}$ .

We refer to Section 5 for the precise definitions of the terms appearing in this theorem. Note that Benjamini, Lyons, and Schramm gave some sufficient conditions for positive speed in [BLS99], and hence Theorem 1.2 can be applied to various models (see Theorem 5.6). For example, every ergodic invariant percolation with a unique infinite cluster satisfies the assumptions in our theorem.

1.2. Outline of the proof

Let us give an overview of the proof of Theorem 1.2. First, the upper bound,

\limsup_{r\to 0}\frac{\log\nu_{\omega}(B(\xi,r))}{\log r}\leq\frac{h}{l}

for $\nu_{\omega}$ -almost every $\xi$ , follows from Kaimanovich’s argument for trees (Theorem 1.4.1 of [Kai98]). The difficulty arises in the proof of the lower bound, i.e.,

\liminf_{r\to 0}\frac{\log\nu_{\omega}(B(\xi,r))}{\log r}\geq\frac{h}{l}

for $\nu_{\omega}$ -almost every $\xi$ . We first prove that, for every $\epsilon>0$ , the set $D_{\epsilon}$ defined by

D_{\epsilon}=\biggl{\{}(\omega,\xi)\in\Omega_{1}\times\partial G\colon\liminf_{r\to 0}\frac{\log\nu_{\omega}(B(\xi,r))}{\log r}\geq\frac{h}{l}-\epsilon\biggr{\}}

is positive with respect to the measure on $\Omega_{1}\times\partial G$ given by $\int_{\Omega_{1}}\delta_{\omega}\otimes\nu_{\omega}d\mu(\omega)$ . This part follows the proof of Theorem 3.3 in [Tan19], where a similar claim was shown for random walks driven by a single measure. Tanaka [Tan19] proved that the harmonic measure determined by the random walk driven by a single measure with finite first moment is exact dimensional and satisfies the dimension formula. In his proof, the ergodicity of the $\Gamma$ -action on $\partial G$ was used to prove that the subset of $\partial G$ defined analogously to $D_{\epsilon}$ is conull, and it completes the proof of the lower bound in his setting.
In our setting, to prove that $D_{\epsilon}$ is conull, the first attempt should be to consider the diagonal action on $\Omega_{1}\times\partial G$ , instead of the boundary action. However, since $\Omega_{1}$ is not $\Gamma$ -invariant, we cannot do this naively. Instead, we define a subrelation of the orbit equivalence relation of the diagonal action, inspired by cluster relations in [Gab05], so that we can prove that the ergodicity of the relation and show that $D_{\epsilon}$ is invariant under the relation. Combining the ergodicity and the invariance with the positivity of $D_{\epsilon}$ , we complete the proof. Such an argument is quite simple but has not appeared in the literature. In fact, in the case of free groups [Kai98], the lower bound can be shown for essentially arbitrary random walk with positive speed in general, and hence such an argument involving the ergodicity on the boundary does not appear.

1.3. Related works and applications

Among hyperbolic groups, cocompact Fuchsian groups have been well studied in the context of percolation theory. They are closely connected to periodic tilings of the hyperbolic plane $\mathbb{H}^{2}$ . Carrasco, Lessa, and Paquette developed the theory of a general class of random walks on metric spaces called distance stationary sequences and applied it to the simple random walks in percolation clusters on cocompact Fuchsian groups in [CLP21]. In particular, they gave an explicit lower bound for the speed of such random walk in terms of the corresponding hyperbolic tiling. Combining our dimension formula with their estimate, we can show the dimension drop of the harmonic measures. This generalizes their Theorem 4 in [CLP21]. More precisely, for the pair of positive integers $P$ and $Q$ with $1/P+1/Q<1/2$ , let $T_{P,Q}$ be the regular tiling of $\mathbb{H}^{2}$ by $P$ -gons with interior angles $2\pi/Q$ and $(\Gamma_{P,Q},S_{P,Q})$ be the pair of the cocompact Fuchsian group and its generating set corresponding to $T_{P,Q}$ . Let $G_{P,Q}$ be the Cayley graph associated with $(\Gamma_{P,Q},S_{P,Q})$ , which is the dual graph obtained from $T_{P,Q}$ . We consider the metric $d_{\mathbb{H}}$ on $G_{P,Q}$ induced from the standard hyperbolic metric of $\mathbb{H}^{2}$ .

Theorem 1.3.

Let $G_{P,Q}$ be the graph defined as above, and $\mu_{p,P,Q}$ denote a supercritical Bernoulli percolation on $G_{P,Q}$ with $p\in[0,1]$ . Then, the harmonic measures determined by $\mu_{p,P,Q}$ are exact dimensional and satisfy the dimension formula with respect to $d_{\mathbb{H}}$ , and the dimension is a constant, denoted by $\delta_{p,P,Q}$ , for $\mu_{p,P,Q}$ -almost every $\omega\in\Omega_{1}$ . Further, we have

\limsup_{Q\to\infty}\delta_{p,P,Q}\leq\frac{1}{2}

uniformly in $P$ .

A natural question arising from our result is about the behavior of the dimension of the harmonic measures determined by the Bernoulli percolation $\mu_{p}$ when the parameter $p$ varies. In [Lal01], Lalley treated the limit sets of percolation clusters on the boundary and proved that its Hausdorff dimension is continuous in the parameter $p$ . Then, it is natural to ask if similar properties also hold for the dimension of the harmonic measures. Our result reduces the continuity of $\dim\nu_{\omega}$ to the continuity of the entropy and the speed. The latter question seems more tractable; we leave it open.
Harmonic measures are also studied in the context of random discretizations of the hyperbolic plane. Angel, Hutchcroft, Nachmias, and Ray studied the simple random walks on unimodular random triangulations of the hyperbolic plane (such as the Poisson-Delaunay triangulations) in [AHNR16], and show that the associated harmonic measures have full support and no atom. Our method applies to this setting, and we will treat it in our next paper.

1.4. Organization of the paper

In Section 2, we review some definitions and basic properties concerning word hyperbolic groups, random walks on them, and Hausdorff dimensions of measures. In Section 3, we show two estimates for general Markov chains with positive speed. In Section 4, we first treat invariant random conductance models with the uniform elliptic condition, where the arguments are simpler than that for invariant percolations. In Section 5, we develop ergodic theory of invariant percolations and use it to prove the exact dimensionality and the dimension formula. In Section 6, we present an application to cocompact Fuchsian groups and hyperbolic tilings. In Section 7, we propose some questions naturally arising from our results. In Appendix A, we give the proof of Theorem 5.5. Although such results are standard and well known to experts, we give the detailed proof for the completeness.

Acknowledgements

The author would like to thank Yoshikata Kida and Ryokichi Tanaka for their supports and helpful comments. This research was supported by FoPM, WINGS Program, the University of Tokyo.

2. Preliminaries

2.1. Geometry of word hyperbolic groups

Let us start with the definition of hyperbolicity in the sense of Gromov [Gro87].

Definition 2.1 ( $\delta$ -hyperbolicity).

Let $(X,d)$ be a proper metric space. For $x,y,z\in X$ , we define the Gromov product of $x,y$ over $z$ by

(x\,|\,y)_{z}=\frac{d(x,z)+d(y,z)-d(x,y)}{2}.

Let $\delta$ be a non-negative number. We say that $(X,d)$ is $\delta$ -hyperbolic if

(x\,|\,y)_{w}\geq\min\{(x\,|\,z)_{w},(y\,|\,z)_{w}\}-\delta

for all $x,y,z,w\in X$ . We say that $(X,d)$ is hyperbolic if it is $\delta$ -hyperbolic for some $\delta\geq 0$ .

We focus on hyperbolic Cayley graphs. Throughout this paper, we always assume that a finite generating set $S$ of a group is symmetric and $1\notin S$ .

Definition 2.2 (Hyperbolic groups).

Let $\Gamma$ be a finitely generated group. We say that $\Gamma$ is hyperbolic if there exists a finite generating set $S$ of $\Gamma$ such that the Cayley graph $G$ associated with $(\Gamma,S)$ is hyperbolic with respect to the graph metric $d_{G}$ . A hyperbolic group $\Gamma$ is called elementary if it is finite or virtually ${\mathbb{Z}}$ . We always take $1$ as the base point of $G$ and define $\lvert g\rvert=d_{G}(1,g)$ for $g\in\Gamma$ .

In the rest of this section, $\Gamma$ denotes a nonelementary hyperbolic group and $G$ denotes its Cayley graph with respect to some generating set $S$ of $\Gamma$ . Note that such $\Gamma$ is nonamenable.

Let us define the boundary of $G$ . For $x,y\in G$ , $(x\,|\,y)$ denotes the Gromov product of $x$ and $y$ over $1$ .

Definition 2.3 (Gromov boundary).

Let $G$ be a hyperbolic Cayley graph endowed with the graph metric. We define the boundary $\partial G$ as follows:

•

The set $\partial G$ is the quotient of the set of geodesic rays starting from $1$ by identifying two rays when they are within a bounded distance.

•

The quasi-metric $\rho$ on $\partial G$ is defined by $\rho(\xi,\xi^{\prime})=\exp{(-(\xi\,|\,\xi^{\prime}))}$ , where $(\xi\,|\,\xi^{\prime})$ is defined by

(\xi\,|\,\xi^{\prime})=\sup\{\liminf_{n,m\to\infty}(x_{n}\,|\,x^{\prime}_{m})\colon x_{n}\in[\xi],x^{\prime}_{m}\in[\xi^{\prime}]\}.

Here, $[\xi]$ denotes the set of geodesic rays from $1$ represent $\xi$ .

Although $\rho$ is not a genuine metric, for every small $\epsilon$ , there exists a genuine metric $\rho_{\epsilon}$ satisfying

C_{\epsilon}^{-1}\rho^{\epsilon}(\xi,\xi^{\prime})\leq\rho_{\epsilon}(\xi,\xi^{\prime})\leq C_{\epsilon}\rho^{\epsilon}(\xi,\xi^{\prime})

where $C_{\epsilon}$ is a constant depending only on $\epsilon$ . The fact that $\rho$ is not a genuine metric has no effect on the argument concerning the Hausdorff dimension. Then we only treat $\rho$ for simplicity.

For the $\Gamma$ -action on $\partial G$ , we have the following estimate.

Lemma 2.4 (Lemma 2.2 in [Tan19]).

Let $g\in\Gamma$ . Then there exists $c>0$ such that

B(g\xi,c^{-1}R)\subset gB(\xi,R)\subset B(g\xi,cR)

for every $R>0$ and $\xi\in\partial G$ .

Let us introduce the shadow of $x\in\Gamma$ on the boundary.

Definition 2.5.

Let $x\in\Gamma$ and $R>0$ . We define the shadow of $(x,R)$ by

S(x,R)=\{\xi\in\partial G\colon(\xi|x)>\lvert x\rvert-R\}.

We have the following comparison between shadows and balls.

Lemma 2.6 (Proposition 2.1 in [BHM11]).

There exist $C>0$ and $R_{0}>0$ such that

B(\xi,C^{-1}e^{-\lvert x\rvert+R})\subset S(x,R)\subset B(\xi,Ce^{-\lvert x\rvert+R})

if $R\geq R_{0}$ and $x\in\Gamma$ is on a geodesic ray from $1$ to $\xi\in\partial G$ .

2.2. Hausdorff dimensions of measures

Definition 2.7 (Hausdorff dimensions of measures).

Let $\nu$ be a Borel probability measure on $\partial G$ . We define the (upper) Hausdorff dimension of $\nu$ by

\dim\nu=\inf\{\dim_{\rho}E\colon\nu(E)=1\},

where $\dim_{\rho}E$ denotes the Hausdorff dimension of $E$ with respect to the quasi-metric $\rho$ .

We have the following characterization of the dimension.

Lemma 2.8.

Let $\nu$ be a Borel probability measure on the boundary $\partial G$ . Then the Hausdorff dimension of $\nu$ can be characterized as follows:

\dim\nu=\underset{\xi\in\partial G}{\nu\text{-}\sup}\liminf_{r\to 0}\frac{\log\nu(B(\xi,r))}{\log r},

where “ $\nu\text{-}\sup$ ” indicates the essential supremum with respect to $\nu$ , and $B(\xi,r)$ is the ball of radius $r$ centered at $\xi$ with respect to the quasi-metric $\rho$ .

Proof.

See Section 1.3 in [Kai98], for example. ∎

Definition 2.9 (Exact dimensionality).

Let $\nu$ be a Borel probability measure on $\partial G$ . We say that $\nu$ is exact dimensional if the limit

\lim_{r\to 0}\frac{\log\nu(B(\xi,r))}{\log r}

exists for $\nu$ -almost every $\xi$ and it is constant $\nu$ -almost everywhere. Note that if $\nu$ is exact dimensional then the above limit is equal to $\dim\nu$ by Lemma 2.8.

2.3. Regular Markov chains on hyperbolic groups

We review some basic properties of nearest neighbor random walks on $G$ . We always assume that Markov chains are nearest neighbor, defined on a hyperbolic Cayley graph $G$ , and starting from $1$ .

Definition 2.10 (Regular Markov chains).

Let $P$ be a nearest neighbor Markov kernel on $G$ and $X$ be the corresponding Markov chain starting from $1$ . Let $\mathbb{P}$ be the distribution of sample paths of $X$ , which is a probability measure on $\Gamma^{{\mathbb{Z}}_{+}}$ . We say that $X$ is regular if there exist reals $h,l\geq 0$ satisfying the following:

•

$\lim_{n\to\infty}\frac{-\log p^{n}(1,x_{n})}{n}=h$ for $\mathbb{P}$ -almost every $x=(x_{n})_{n\in{\mathbb{Z}}_{+}}$ . Here $p^{n}$ denotes the $n$ -step transition probability of $P$ .
•

$\lim_{n\to\infty}\frac{\lvert x_{n}\rvert}{n}=l$ for $\mathbb{P}$ -almost every $x=(x_{n})_{n\in{\mathbb{Z}}_{+}}$ .

The limits $h,l$ are called the entropy and the speed of $X$ , respectively.

We will show that if $X$ is a Markov chain determined by invariant random conductance model or invariant percolation then it is regular and the entropy and the speed are both positive. See Theorem 4.4 and Theorem 5.6 for precise statements.

The following result is fundamental in the study of random walks on hyperbolic groups, shown by Kaimanovich in [Kai00].

Theorem 2.11 (Section 7 of [Kai00]).

Let $X$ be a regular Markov chain on a non-elementary hyperbolic group $\Gamma$ . Then for $\mathbb{P}$ -almost every $x=(x_{n})_{n\in{\mathbb{Z}}_{+}}$ , it converges to a point in $\partial G$ . More precisely, for $\mathbb{P}$ -almost every sample path $x=(x_{n})_{n\in{\mathbb{Z}}_{+}}$ , there exists a unit speed geodesic ray $\xi_{x}$ such that

\lim_{n\to\infty}\frac{1}{n}d_{G}(x_{n},\xi_{x}(ln))=0,

where $l$ is the speed of $X$ .

This property is often called the geodesic tracking property. By this theorem, the boundary map $(x_{n})_{n\in{\mathbb{Z}}_{+}}\mapsto\xi_{x}\in\partial G$ is well-defined for almost every sample path $x=(x_{n})_{n\in{\mathbb{Z}}_{+}}$ and written as $\operatorname{bnd}\colon\Gamma^{{\mathbb{Z}}_{+}}\to\partial G$ . The harmonic measure $\nu$ of $X$ is defined as $\operatorname{bnd}_{*}\mathbb{P}$ .

3. General Estimates for Dimensions

The goal of this section is to prove the following two general estimates. Note that both of them have already been shown for random walks driven by a single measure on hyperbolic groups in [LP07] and [Tan19], respectively. Let $\Gamma$ denote a nonelementary $\delta$ -hyperbolic group and $G$ denote its Cayley graph.

Theorem 3.1.

Let $X$ be a regular and nearest neighbor Markov chain having positive speed on $G$ . Then, for $\nu$ -almost every $\xi$ , we have

\limsup_{r\to 0}\frac{\log\nu(B(\xi,r))}{\log r}\leq\frac{h}{l},

where $h$ and $l$ denote the entropy and the speed of $X$ , respectively. In particular, $\dim\nu\leq\frac{h}{l}$ .

Theorem 3.2.

Let $X$ be a regular and nearest neighbor Markov chain having positive speed on $G$ . Then, for every $\epsilon>0$ , the following subset $D_{\epsilon}$ is $\nu$ -positive:

D_{\epsilon}:=\biggl{\{}\xi\in\partial G:\liminf_{r\to 0}\frac{\log\nu(B(\xi,r))}{\log r}\geq\frac{h}{l}-\epsilon\biggr{\}}.

Proof of Theorem 3.1..

We follow Section 2 in [LP07]. Let $\mathbb{P}$ denote the probability on $\Gamma^{{\mathbb{Z}}_{+}}$ defined as the distribution of $X$ . For $\epsilon>0$ and $N\in{\mathbb{N}}$ , we define the set

A_{\epsilon,N}=\{x\in\Gamma^{{\mathbb{Z}}_{+}}\colon(x_{n+1}\,|\,x_{n})\geq(l-\epsilon)n\,\,\text{and}\,\,p^{n}(1,x_{n})\geq e^{-(h+\epsilon)n}\,\,\text{for every}\,\,n\geq N\}.

Then, for each $\epsilon$ , there exists $N_{\epsilon}$ such that $\mathbb{P}(A_{\epsilon,N_{\epsilon}})\geq 1-\epsilon$ by the regularity of $X$ . Let $A_{\epsilon}$ denote this $A_{\epsilon,N_{\epsilon}}$ . We first show that the limit

\lim_{n\to\infty}\mathbb{P}(A_{\epsilon}\,|\,\{x_{n}=z_{n}\})

exists for almost every $z\in A_{\epsilon}$ . This follows from the Markov property of $X$ . Indeed, for $n>N_{\epsilon}$ , we have

	$\displaystyle\lim_{n\to\infty}\mathbb{P}(A_{\epsilon}\,\|\,\{x_{n}=z_{n}\})$	$\displaystyle=\lim_{n\to\infty}\mathbb{P}(A_{\epsilon}\,\|\,\{x_{m}=z_{m}\,\,\text{for every}\,\,m\geq n\})$
		$\displaystyle=\mathbb{P}(A_{\epsilon}\,\|\,\textbf{tail}(z))$

by the martingale convergence theorem, where tail denotes the projection to the tail boundary. We define $A^{\prime}_{\epsilon}=\{z\in A_{\epsilon}\colon\mathbb{P}(A_{\epsilon}\,|\,\textbf{tail}(z))>\epsilon\}$ . Then, we have

	$\displaystyle 1-\epsilon$	$\displaystyle\leq\mathbb{P}(A_{\epsilon})=\int\mathbb{P}(A_{\epsilon}\,\|\,\textbf{tail}(x))d\mathbb{P}(x)$
		$\displaystyle=\int_{A^{\prime}_{\epsilon}}\,\mathbb{P}(A_{\epsilon}\,\|\,\textbf{tail}(x))d\mathbb{P}(x)+\int_{(A^{\prime}_{\epsilon})^{c}}\,\mathbb{P}(A_{\epsilon}\,\|\,\textbf{tail}(x))d\mathbb{P}(x)$
		$\displaystyle\leq\mathbb{P}(A^{\prime}_{\epsilon})+\epsilon$

and hence $\mathbb{P}(A^{\prime}_{\epsilon})\geq 1-2\epsilon$ . Recall that for $z\in A_{\epsilon}$ , there exists $C>0$ such that

A_{\epsilon}\cap\{x_{n}=z_{n}\}\subset\{x\in\Gamma^{{\mathbb{Z}}_{+}}\colon\xi_{x}\in B(\xi_{z},Ce^{-(l-\epsilon)n})\}

for every $n\geq N_{\epsilon}$ . Then, for $z\in A^{\prime}_{\epsilon}$ , we have

	$\displaystyle\limsup_{r\to 0}\frac{\log\nu(B(\xi_{z},r))}{\log r}$	$\displaystyle=\limsup_{n\to\infty}\frac{\log\nu(B(\xi_{z},Ce^{-(l-\epsilon)n}))}{-(l-\epsilon)n}\leq\limsup_{n\to\infty}\frac{\log\mathbb{P}(A^{\prime}_{\epsilon}\cap\{x_{n}=z_{n}\})}{-(l-\epsilon)n}$
		$\displaystyle=\limsup_{n\to\infty}\frac{\log\mathbb{P}(\{x_{n}=z_{n}\})}{-(l-\epsilon)n}=\limsup_{n\to\infty}\frac{\log p^{n}(1,z_{n})}{-(l-\epsilon)n}$
		$\displaystyle\leq\frac{h+\epsilon}{l-\epsilon}$

by combining the above estimates. Since $\epsilon$ can be taken arbitrarily and $\mathbb{P}(A^{\prime}_{\epsilon})>1-2\epsilon$ , this implies

\limsup_{r\to 0}\frac{\log\nu(B(\xi,r))}{\log r}\leq\frac{h}{l}

for $\nu$ -almost every $\xi$ . We also have $\dim\nu\leq\frac{h}{l}$ by Lemma 2.8. ∎

Next, we give the proof of Theorem 3.2, following Theorem 3.3 in [Tan19]. In the proof, we consider the conditional measures associated with the factor map $\operatorname{bnd}\colon(\Gamma^{{\mathbb{Z}}_{+}},\mathbb{P})\to(\partial G,\nu)$ . More explicitly, that is a family of probability measures $(\mathbb{P}^{\xi})_{\xi\in\partial G}$ on $\Gamma^{{\mathbb{Z}}_{+}}$ such that

\mathbb{P}=\int_{\partial G}\mathbb{P}^{\xi}\,\,d\nu.

Proof of Theorem 3.2..

It is enough to prove that there exists a $\nu$ -positive set $F_{\epsilon}$ such that

\liminf_{r\to 0}\frac{\log\nu(F_{\epsilon}\cap B(\xi,r))}{\log r}\geq\frac{h}{l}-\epsilon

for $\nu$ -almost every $\xi\in\partial G$ . Indeed, for such $F_{\epsilon}$ , we have $\dim\nu\geq\dim\nu|_{F_{\epsilon}}\geq\frac{h}{l}-\epsilon$ by the definition of Hausdorff dimensions and Lemma 2.8. Then, by Lemma 2.8, it implies that $D_{\epsilon}$ is positive. In the rest of the proof, we construct such $F_{\epsilon}$ . First, we define $A_{\epsilon,N}$ as follows:

A_{\epsilon,N}=\{x\in\Gamma^{{\mathbb{Z}}_{+}}\colon d(x_{n},\xi_{x}(ln))\leq\epsilon n\,\,\text{and}\,\,p^{n}(1,x_{n})\leq e^{-(h-\epsilon)n}\,\,\text{for every}\,\,n\geq N\}

for $\epsilon>0$ and $N\in{\mathbb{N}}$ . Then, for every $\epsilon>0$ , there exists $N_{\epsilon}$ such that $\mathbb{P}(A_{\epsilon,N_{\epsilon}})\geq 1-\epsilon$ by regularity of $X$ and Theorem 2.11. Let $A_{\epsilon}=A_{\epsilon,N_{\epsilon}}$ and $F_{\epsilon}=\{\xi\,\colon\,\mathbb{P}^{\xi}(A_{\epsilon})\geq\epsilon\}$ . Then we can estimate the size of $F_{\epsilon}$ as follows:

	$\displaystyle 1-\epsilon$	$\displaystyle\leq\mathbb{P}(A_{\epsilon})=\int\mathbb{P}^{\xi}(A_{\epsilon})\,d\nu(\xi)$
		$\displaystyle=\int_{F_{\epsilon}}\,\mathbb{P}^{\xi}(A_{\epsilon})\,d\nu(\xi)+\int_{F^{c}_{\epsilon}}\,\mathbb{P}^{\xi}(A_{\epsilon})\,d\nu(\xi)$
		$\displaystyle\leq\nu(F_{\epsilon})+\epsilon$

and hence $\nu(F_{\epsilon})\geq 1-2\epsilon$ . Let $N\geq N_{\epsilon}$ and $R>\max\{4\delta,R_{0}\}$ , where $R_{0}$ is the constant in Lemma 2.6. For $z\in A_{\epsilon,N}$ , we define $y_{n}$ as $\xi_{z}(ln)$ . Then, by Lemma 3.6 in [Tan19] (note that the assumption $R>4\delta$ is used here), there exists $C>0$ such that for all $n\geq N$ and $x\in A_{\epsilon}$ , $x_{n}$ belongs to the ball $B(y_{n},2\epsilon n+C)$ if $\xi_{x}\in S(y_{n},R)$ . Therefore, we have

	$\displaystyle\mathbb{P}(\xi_{x}\in F_{\epsilon}\cap S(y_{n},R))$	$\displaystyle\leq\mathbb{P}(A_{\epsilon}\cap\{x_{n}\in B(y_{n},2\epsilon n+C)\})+\mathbb{P}(A^{c}_{\epsilon}\cap\{\xi_{x}\in F_{\epsilon}\cap S(y_{n},R)\})$
		$\displaystyle\leq\mathbb{P}(A_{\epsilon}\cap\{x_{n}\in B(y_{n},2\epsilon n+C)\})+(1-\epsilon)\nu(F_{\epsilon}\cap S(y_{n},R))$

for every $n\geq N$ . The last inequality follows from the construction of $F_{\epsilon}$ . Then we have

	$\displaystyle\epsilon\nu(F_{\epsilon}\cap S(y_{n},R))$	$\displaystyle\leq\mathbb{P}(A_{\epsilon}\cap\{x_{n}\in B(y_{n},2\epsilon n+C)\})$
		$\displaystyle\leq d^{2\epsilon n+C}\times e^{-(h-\epsilon)n},$

where $d$ is the degree of $G$ . From this inequality, we obtain

\displaystyle\liminf_{n\to\infty}\frac{\log\nu(F_{\epsilon}\cap S(y_{n},R))}{-ln}\geq\liminf_{n\to\infty}\frac{-\log\epsilon-(h-\epsilon)n+(2\epsilon n+C)\log d}{-ln}=\frac{h-(1+2\log d)\epsilon}{l}

Further, by applying Lemma 2.6 to $y_{n}$ , which is on a geodesic ray from $1$ to $\xi_{z}$ , we have

S(y_{n},R)\subset B(\xi_{z},Ce^{-ln+R})

and hence

	$\displaystyle\liminf_{r\to 0}\frac{\log\nu(F_{\epsilon}\cap B(\xi_{z},r))}{\log r}$	$\displaystyle=\liminf_{n\to\infty}\frac{\log\nu(F_{\epsilon}\cap B(\xi_{z},Ce^{-ln+R}))}{\log(Ce^{-ln+R})}=\liminf_{n\to\infty}\frac{\log\nu(F_{\epsilon}\cap B(\xi_{z},Ce^{-ln+R}))}{-ln}$
		$\displaystyle\geq\liminf_{n\to\infty}\frac{\log\nu(F_{\epsilon}\cap S(y_{n},R))}{-ln}$

since $C$ and $R$ are positive constants.
Finally, for $z\in A_{\epsilon,N}$ , we obtain

	$\displaystyle\liminf_{r\to 0}\frac{\log\nu(F_{\epsilon}\cap B(\xi_{z},r))}{\log r}$	$\displaystyle\geq\liminf_{n\to\infty}\frac{\log\nu(F_{\epsilon}\cap S(y_{n},R))}{-ln}$
		$\displaystyle\geq\frac{h}{l}-\frac{(1+2\log d)\epsilon}{l}$

from the above estimates. Since the sequence $(A_{\epsilon,N})_{N\geq N_{\epsilon}}$ gives an exhaustion of $\Gamma^{{\mathbb{Z}}_{+}}$ modulo null sets, we conclude that $F_{\epsilon}$ satisfies the desired property. ∎

4. Invariant Random Conductance Models

In this section, we consider uniformly elliptic random conductance models. They can be seen as bounded random perturbations of $G$ , and the proof of the exact dimensionality is simpler than the case of percolation clusters. Readers interested in percolation can skip this section.

4.1. Ergodic Theory of Random Conductance Models

Definition 4.1 ( $\Gamma$ -invariant random conductance models).

Let $\mathcal{C}$ be the space of conductances ${\mathbb{R}}_{>0}^{E}$ , where $E$ denotes the edge set of $G$ . Note that $\Gamma$ acts on $\mathcal{C}$ by translations. A $\Gamma$ -invariant random conductance model is a $\Gamma$ -invariant probability measure on $\mathcal{C}$ . We say that a $\Gamma$ -invariant random conductance model $\mu$ is ergodic if it is ergodic under the $\Gamma$ -action. Each element $\omega$ of $\mathcal{C}$ determines a nearest neighbor Markov kernel on $G$ as follows:

p_{\omega}(g,gs)=\frac{\omega(g,gs)}{\sum_{s^{\prime}\in S}\omega(g,gs^{\prime})}

for $g\in\Gamma$ and $s\in S$ , and other transition probabilities are zero. This kernel is denoted by $P_{\omega}$ and the corresponding Markov chain starting from $1$ is denoted by $X_{\omega}$ . We call $\mu$ uniformly elliptic if $\text{supp}(\mu)$ is contained in $(\alpha,1/\alpha)^{E}$ for some $\alpha\in(0,1)$ .

Definition 4.2 (Path bundles).

Let $\mu$ be a $\Gamma$ -invariant uniformly elliptic random conductance model on $G$ . We define the weighted version $\mu^{\prime}$ of $\mu$ by

d\mu^{\prime}(\omega)=\frac{\sum_{s\in S}\omega(1,s)}{D}d\mu(\omega),

where $D$ is the expectation of the sum of conductances at $1$ . We define the path bundle $\Pi$ by $\mathcal{C}\times\Gamma^{{\mathbb{Z}}_{+}}$ as a Borel space, and we endow $\Pi$ with the probability measure $\lambda$ defined by

\lambda=\int_{\mathcal{C}}\delta_{\omega}\otimes\mathbb{P}_{\omega}d\mu^{\prime}(\omega),

where $\mathbb{P}_{\omega}$ denotes the distribution of sample paths of $X_{\omega}$ , which is a probability measure on $\Gamma^{{\mathbb{Z}}_{+}}$ . Further, we define the shift map $T$ on $\Pi$ by $(\omega,(x_{n})_{n\in{\mathbb{Z}}_{+}})\mapsto(x_{1}^{-1}\omega,(x_{1}^{-1}x_{n+1})_{n\in{\mathbb{Z}}_{+}})$ .

The following result is crucial for our ergodic-theoretic approach. Note that this type of results are quite standard in the context of random walks in random environments.

Theorem 4.3.

Let $\mu$ be an ergodic $\Gamma$ -invariant random conductance model on $G$ . Then the system $(\Pi,\lambda,T)$ is an ergodic probability-measure-preserving system.

For the proof, we refer to Appendix A.

This theorem can be used to deduce that $X_{\omega}$ is regular for $\mu$ -almost every $\omega$ . Further, we can show that the entropy and the speed are positive and constant for $\mu$ -almost every $\omega$ .

Theorem 4.4.

Let $\mu$ be an ergodic $\Gamma$ -invariant uniformly elliptic random conductance model. Then for $\mu$ -almost every $\omega\in\mathcal{C}$ , the random walk $X_{\omega}$ is regular for $\mu$ -almost every $\omega\in\mathcal{C}$ , and its entropy and speed do not depend on $\omega$ and are positive.

Proof.

First, we show that for $\mu$ -almost every $\omega$ , $X_{\omega}$ is regular and its entropy and speed are constant. For $n\in{\mathbb{Z}}_{+}$ , we define functions on $\Pi$ as follows:

•

$\phi_{n}(\omega,x)=\lvert x_{n}\rvert$ ,
•

$\psi_{n}(\omega,x)=-\log p^{n}_{\omega}(1,x_{n})$ .

Then we have

\displaystyle\phi_{m+n}(\omega,x)

\displaystyle=\lvert x_{m+n}\rvert\leq\lvert x_{m}\rvert+\lvert x_{m}^{-1}x_{n}\rvert=\phi_{m}(\omega,x)+\phi_{n}(T^{m}(\omega,x))

and

	$\displaystyle\psi_{m+n}(\omega,x)$	$\displaystyle=-\log p^{m+n}_{\omega}(1,x_{m+n})$
		$\displaystyle\leq-\log p^{m}_{\omega}(1,x_{m})p^{n}_{\omega}(x_{m},x_{m+n})$
		$\displaystyle=-\log p^{m}_{\omega}(1,x_{m})-\log p^{n}_{x_{m}^{-1}\omega}(1,x_{m}^{-1}x_{m+n})$
		$\displaystyle=\psi_{m}(\omega,x)+\psi_{n}(T^{m}(\omega,x)).$

Therefore, we can apply Kingman’s ergodic theorem to $\phi_{n}$ and $\psi_{n}$ , and hence for $\mu$ -almost every $\omega$ , $X_{\omega}$ is regular and its entropy and speed are constant. We show that they are positive in the rest of the proof. It is known that the entropy is positive if and only if the speed is positive (see Proposition 3.6 in [BC12] for example). Therefore it is enough to show that the speed is positive. For a subset $B$ of the vertices of $G$ , we define its edge-boundary $\partial_{E}(B)$ by

\partial_{E}(B)=\{e\in E\colon e\,\,\text{connects a vertex in}\,\,B\,\,\text{and a vertex in}\,\,B^{c}\}.

Then, for $\omega\in\mathcal{C}$ and every nonempty finite subset $K$ of the vertices of $G$ , we have

\frac{\sum_{e\in\partial_{E}(K)}\omega(e)}{\lvert K\rvert}\geq\alpha\frac{\lvert\partial_{E}(K)\rvert}{\lvert K\rvert}

where $\alpha>0$ is the lower bound for $\omega(e)$ , guaranteed by the uniform ellipticity. Since $G$ is nonamenable, this implies that the network determined by $\omega$ is nonamenable. Then, the claim follows from the fact that the random walk on a nonamenable network with exponential growth (note that $G$ has exponential growth since $\Gamma$ is nonamenable) has positive speed (see Section 6.2 in [LP16]). ∎

Then, we can consider the harmonic measures (Theorem 2.11) and the ray bundle in the above setting, as follows.

Definition 4.5 (Ray bundles).

The ray bundle $\Xi$ is $\mathcal{C}\times\partial G$ as a Borel space, and we consider the probability measure $\eta$ on $\Xi$ by

\eta=\int_{\mathcal{C}}\delta_{\omega}\otimes\nu_{\omega}d\mu^{\prime}(\omega),

where $\nu_{\omega}$ denotes the harmonic measure associated with $X_{\omega}$ for $\mu$ -almost every $\omega$ . Note that $\eta$ is equal to the pushforward measure $(\operatorname{id}\times\operatorname{bnd})_{*}\lambda$ .

Lemma 4.6.

The diagonal action of $\Gamma$ on $\Xi=\mathcal{C}\times\partial G$ is ergodic with respect to $\eta$ .

Proof.

Let $A$ be a $\Gamma$ -invariant subset of $\Xi$ . Since $\eta$ is equal to the pushforward measure $(\operatorname{id}\times\operatorname{bnd})_{*}\lambda$ , it is enough to prove that the inverse image of $A$ under $\operatorname{id}\times\operatorname{bnd}$ is $\lambda$ -null or $\lambda$ -conull. Note that the $\Gamma$ -invariance of $A$ implies the $T$ -invariance of $(\operatorname{id}\times\operatorname{bnd})^{-1}(A)$ by the definition of $T$ . Since $T$ is ergodic with respect to $\lambda$ by the Theorem 4.3, it implies the claim. ∎

4.2. The Dimension Formula for Invariant Random Conductance Models

In this subsection, we prove the main theorem for random conductance models as above. First, we have the following two results:

Theorem 4.7.

Let $\mu$ be an ergodic $\Gamma$ -invariant uniformly elliptic random conductance model on $G$ . Then, for $\mu$ -almost every $\omega\in\mathcal{C}$ and $\nu_{\omega}$ -almost every $\xi$ , we have

\limsup_{r\to 0}\frac{\log\nu_{\omega}(B(\xi,r))}{\log r}\leq\frac{h}{l},

where $h$ and $l$ denote the entropy and the speed of $X_{\omega}$ , respectively. In particular, $\dim\nu_{\omega}\leq\frac{h}{l}$ .

Proof.

This follows from Theorem 3.1 and Theorem 4.4. ∎

Theorem 4.8.

Let $\mu$ be an ergodic $\Gamma$ -invariant uniformly elliptic random conductance model on $G$ . Then, for $\epsilon>0$ and $\mu$ -almost every $\omega\in\mathcal{C}$ , the set $D_{\epsilon}$ defined by

D_{\epsilon}=\biggl{\{}(\omega,\xi)\in\mathcal{C}\times\partial G:\liminf_{r\to 0}\frac{\log\nu_{\omega}(B(\xi,r))}{\log r}\geq\frac{h}{l}-\epsilon\biggr{\}}

is $\eta$ -positive.

Proof.

This follows from Theorem 3.2 and Theorem 4.4. ∎

Combining these theorems with the ergodicity established in the previous section, we have the following result.

Theorem 4.9.

Let $\mu$ be an ergodic $\Gamma$ -invariant uniformly elliptic random conductance model on $G$ . Then, for $\mu$ -almost every $\omega\in\mathcal{C}$ , letting $\nu_{\omega}$ be the harmonic measure on $\partial G$ determined by $X_{\omega}$ , we have

\dim\nu_{\omega}=\lim_{r\to 0}\frac{\log\nu_{\omega}(B(\xi,r))}{\log r}=\frac{h}{l}

for $\nu_{\omega}$ -almost every $\xi$ , where $h$ and $l$ denote the entropy and the speed of $X_{\omega}$ , respectively. In particular, $\dim\nu_{\omega}$ is positive and constant for $\mu$ -almost every $\omega\in\mathcal{C}$ .

Proof.

We have already shown that the upper bound in Theorem 4.7. The lower bound can be deduced from Theorem 4.8 as follows. We want to show that $D_{\epsilon}$ is $\eta$ -conull for every $\epsilon>0$ . It is enough to prove that $D_{\epsilon}$ is invariant under the diagonal action of $\Gamma$ since the diagonal action of $\Gamma$ is ergodic with respect to $\eta$ (Lemma 4.6) and $D_{\epsilon}$ is $\eta$ -positive (Theorem 4.8). Let $(\omega,\xi)\in D_{\epsilon}$ and $g\in\Gamma$ . For $g\in\Gamma$ , let $\mathbb{P}_{\omega,g}$ denote the distribution of sample paths starting from $g$ following $P_{\omega}$ . Then, for every Borel subset $A\subset\partial G$ , we have

	$\displaystyle(g_{*}\nu_{\omega})(A)$	$\displaystyle=\mathbb{P}_{\omega}(\{x\in\Gamma^{{\mathbb{Z}}_{+}}\colon\xi_{x}\in g^{-1}A\})\geq\mathbb{P}_{\omega}(\bigcup_{n\in{\mathbb{Z}}_{+}}\{x_{n}=g^{-1}\}\cap\{x\in\Gamma^{{\mathbb{Z}}_{+}}\colon\xi_{x}\in g^{-1}A\})$
		$\displaystyle=\mathbb{P}_{\omega}(\bigcup_{n\in{\mathbb{Z}}_{+}}\{x_{n}=g^{-1}\})\mathbb{P}_{\omega,g^{-1}}(\{x\in\Gamma^{{\mathbb{Z}}_{+}}\colon\xi_{x}\in g^{-1}A\})$
		$\displaystyle=\mathbb{P}_{\omega}(\bigcup_{n\in{\mathbb{Z}}_{+}}\{x_{n}=g^{-1}\})\mathbb{P}_{g\omega}(\{x\in\Gamma^{{\mathbb{Z}}_{+}}\colon\xi_{x}\in A\})=\mathbb{P}_{\omega}(\bigcup_{n\in{\mathbb{Z}}_{+}}\{x_{n}=g^{-1}\})\nu_{g\omega}(A),$

where we have used the strong Markov property of $X_{\omega}$ in the second equality. The third equality follows from the equality $\mathbb{P}_{g\omega}=g_{*}\mathbb{P}_{\omega,g^{-1}}$ . Further, by Lemma 2.4, we have $\nu_{g\omega}(B(g\xi,r))\leq\nu_{g\omega}(gB(\xi,c_{g}r))$ , where $c_{g}$ is a constant depending only on $g$ . Using these estimates, we have

	$\displaystyle\liminf_{r\to 0}\frac{\log\nu_{g\omega}(B(g\xi,r))}{\log r}$	$\displaystyle\geq\liminf_{r\to 0}\frac{\log(g_{*}\nu_{\omega})(gB(\xi,c_{g}r))-\log\mathbb{P}_{\omega}(\cup_{n\in{\mathbb{Z}}_{+}}\{x_{n}=g^{-1}\})}{\log r}$
		$\displaystyle=\liminf_{r\to 0}\frac{\log\nu_{\omega}(B(\xi,c_{g}r))}{\log r}=\liminf_{r\to 0}\frac{\log\nu_{\omega}(B(\xi,r))}{\log r}$
		$\displaystyle\geq\frac{h}{l}-\epsilon,$

and hence $(g\omega,g\xi)\in D_{\epsilon}$ , as required. ∎

5. Invariant Percolations

In this section, we consider invariant percolations. The arguments in this section are almost parallel to the case of random conductance models. However, we need some modifications since $\Gamma$ does not act on the space of environments $\Omega_{1}$ in this case. Throughout this section, $\Gamma$ denotes a nonelementary hyperbolic group and $G$ denotes its Cayley graph with respect to some generating set $S$ of $\Gamma$ , and $E$ denotes the edge set of $G$ .

5.1. Ergodic Theory of Invariant Percolations

Definition 5.1 ( $\Gamma$ -invariant percolations).

A $\Gamma$ -invariant percolation on $G$ is a $\Gamma$ -invariant probability measure on $\{0,1\}^{E}$ . Note that $\Gamma$ acts on $\{0,1\}^{E}$ by translations. We say that a $\Gamma$ -invariant percolation $\mu$ is ergodic if it is ergodic under the $\Gamma$ -action. We identify an element $\omega\in\{0,1\}^{E}$ with a spanning subgraph of $G$ such that $e\in\omega$ if and only if $\omega(e)=1$ , and denote by $C_{\omega}(1)$ the connected component (often called cluster) of $\omega$ containing $1$ . We define $\Omega_{1}$ by

\Omega_{1}:=\{\omega\in\{0,1\}^{E}\colon C_{\omega}(1)\,\,\text{is infinite}\}.

Note that $\Omega_{1}$ is not $\Gamma$ -invariant. In the rest of this section, we always assume that $\mu(\Omega_{1})>0$ . Note that by the $\Gamma$ -invariance of $\mu$ , if $\omega$ has an infinite cluster with $\mu$ -positive probability, then $\mu(\Omega_{1})>0$ . Each $\omega\in\Omega_{1}$ gives a Markov kernel $P_{\omega}$ defined on $C_{\omega}(1)$ as follows:

p_{\omega}(g,gs)=\frac{1}{\deg_{C_{\omega}(1)}(g)}

if $g,gs\in C_{\omega}(1)$ and $s\in S$ . Since $\omega\in\Omega_{1}$ , each $g\in C_{\omega}(1)$ has at least one neighbor in $C_{\omega}(1)$ , and hence the denominator is not zero. Let $X_{\omega}$ denote the Markov chain starting from $1$ determined by $P_{\omega}$ .

Let us introduce cluster relations and indistinguishability, following [Gab05] and [GL09]. They can be used as substitutes for the $\Gamma$ -action on the space of environments and its ergodicity, respectively.

Definition 5.2 (Cluster relations and indistinguishability).

Let $\mu$ be an ergodic $\Gamma$ -invariant percolation on $G$ satisfying $\mu(\Omega_{1})>0$ . We define the cluster relation on $\{0,1\}^{E}$ by

\mathcal{R}^{cl}=\{(\omega,g\omega)\in\{0,1\}^{E}\times\{0,1\}^{E}\colon g^{-1}\in C_{\omega}(1)\}.

This is actually an equivalence relation on $\{0,1\}^{E}$ . Note that for $\omega\in\{0,1\}^{E}$ , the $\mathcal{R}^{cl}$ -class containing $\omega$ can be identified with the vertices of $C_{\omega}(1)$ via $g\omega\mapsto g^{-1}$ . We say that $\mu$ has indistinguishable infinite clusters if the restricted relation $\mathcal{R}^{cl}|_{\Omega_{1}}$ is ergodic with respect to $\mu$ . Recall that a countable Borel equivalence relation $\mathcal{R}$ on a standard probability space $(X,\nu)$ is called ergodic if every Borel subset $A\subset X$ satisfying the equation

A=\{x\in X\colon\exists x^{\prime}\in A,(x,x^{\prime})\in\mathcal{R}\}

is $\nu$ -null or $\nu$ -conull. Note that many invariant percolation models (including supercritical Bernoulli percolations) are known to have indistinguishable infinite clusters [LS99].

Remark 5.3.

We have defined the indistinguishability using cluster relations. It is straightforward to check that this definition is equivalent to the original definition in Section 3 of [LS99] under the ergodicity of $\mu$ . See Proposition 5 in [GL09] for details.

Definition 5.4 (Path bundles).

Let $\mu$ be a $\Gamma$ -invariant percolation on $G$ . We define the weighted version $\mu^{\prime}$ of $\mu$ on $\Omega_{1}$ by

d\mu^{\prime}(\omega)=\frac{\deg_{C_{\omega}(1)}(1)}{D}d\mu(\omega).

where $D$ is the expected degree at $1$ on $\Omega_{1}$ . The path bundle $\Pi$ associated with $\mu$ is $\Omega_{1}\times\Gamma^{{\mathbb{Z}}_{+}}$ as a Borel space, and we endow $\Pi$ with the probability measure $\lambda$ defined by

\lambda=\int_{\Omega_{1}}\delta_{\omega}\otimes\mathbb{P}_{\omega}d\mu^{\prime}(\omega),

where $\mathbb{P}_{\omega}$ denotes the distribution of sample paths of $X_{\omega}$ , which is a probability measure on $\Gamma^{{\mathbb{Z}}_{+}}$ . Further, the shift map $T$ on $\Pi$ is defined by $(\omega,(x_{n})_{n\in{\mathbb{Z}}_{+}})\mapsto(x_{1}^{-1}\omega,(x_{1}^{-1}x_{n+1})_{n\in{\mathbb{Z}}_{+}})$ .

The following theorem is crucial for our study.

Theorem 5.5.

Let $\mu$ be an ergodic $\Gamma$ -invariant percolation on $G$ with indistinguishable infinite clusters. Then the system $(\Pi,\lambda,T)$ is ergodic and measure-preserving.

Note that this type of result is well known in the context of random walks in random environments (see [BB07], for example). We refer to Appendix A for the proof.
This theorem implies that the speed and the entropy of the simple random walks on percolation clusters exist and they are constant $\mu$ -almost surely. Note that they may be zero in some invariant percolations like wired uniform spanning forests (see Remark 4.5 in [BLS99]). However, many interesting examples are known to have positive entropy and speed as indicated in the following result due to Benjamini, Lyons, and Schramm.

Theorem 5.6 (Theorem 4.4 in [BLS99]).

Let $G$ be a Cayley graph of a nonamenable group and $\mu$ be an ergodic $\Gamma$ -invariant percolation on $G$ with indistinguishable infinite clusters. Assume one of the following conditions:

•

$\mu$ is a supercritical Bernoulli percolation.
•

$\mu$ -almost every $\omega$ has a unique infinite cluster.
•

$\mu$ -almost every $\omega$ has a cluster with at least three ends.
•

The expected degree at $1$ on $\Omega_{1}$ is larger than $d-i(G)$ , where $i(G)$ is the isoperimetric constant of $G$ .

Then for $\mu$ -almost every $\omega\in\Omega_{1}$ , the simple random walk $X_{\omega}$ on $C_{\omega}(1)$ is regular and its entropy and speed do not depend on $\omega$ and are positive.

In such cases, we can consider the corresponding harmonic measures by Theorem 2.11 and define the ray bundle.

Definition 5.7 (Ray bundles).

Let $\mu$ be an ergodic $\Gamma$ -invariant percolation on $G$ with indistinguishable infinite clusters. Assume that the corresponding simple random walk has positive speed, i.e., the simple random walk $X_{\omega}$ on $C_{\omega}(1)$ has positive speed for $\mu$ -almost every $\omega\in\Omega_{1}$ . Note that the simple random walk $X_{\omega}$ determines the corresponding harmonic measure $\nu_{\omega}$ for $\mu$ -almost every $\omega\in\Omega_{1}$ by Theorem 2.11. The ray bundle $\Xi$ associated with $\mu$ is $\Omega_{1}\times\partial G$ as a Borel space, and we endow $\Xi$ with the probability measure $\eta$ defined by

\eta=\int_{\Omega_{1}}\delta_{\omega}\otimes\nu_{\omega}d\mu^{\prime}(\omega).

Next, we define an equivalence relation which substitutes for the diagonal action of $\Gamma$ on the ray bundle and show its ergodicity with respect to $\eta$ .

Lemma 5.8.

Let $\mu$ be an ergodic $\Gamma$ -invariant percolation on $G$ with indistinguishable infinite clusters. Assume that the corresponding simple random walk has positive speed. Then, the following equivalence relation $\mathcal{R}_{1}$ on $\Xi$ is ergodic with respect to $\eta$ :

\mathcal{R}_{1}=\{((\omega,\xi),(g\omega,g\xi))\in\Xi^{2}\colon(\omega,g\omega)\in\mathcal{R}^{cl}|_{\Omega_{1}}\}.

Proof.

Let $A$ be an $\mathcal{R}_{1}$ -invariant subset of $\Xi=\Omega_{1}\times\partial G$ . Then the inverse image of $A$ under $\operatorname{id}\times\operatorname{bnd}$ is invariant under the shift map $T$ . Since $T$ is ergodic with respect to $\lambda$ (Theorem 5.5) and $(\operatorname{id}\times\operatorname{bnd})_{*}\lambda=\eta$ , it implies that $A$ is null or conull. ∎

5.2. The Dimension Formula for Invariant Percolations

In this subsection, we prove the main theorem of this paper.

Theorem 5.9.

Let $\mu$ be an ergodic $\Gamma$ -invariant percolation with indistinguishable infinite clusters. Assume that the corresponding simple random walk has positive speed. Then, for $\mu$ -almost every $\omega\in\Omega_{1}$ , letting $\nu_{\omega}$ be the harmonic measure on $\partial G$ determined by $X_{\omega}$ , we have

\dim\nu_{\omega}=\lim_{r\to 0}\frac{\log\nu_{\omega}(B(\xi,r))}{\log r}=\frac{h}{l}

for $\nu_{\omega}$ -almost every $\xi\in\partial G$ , where $h$ and $l$ denote the entropy and the speed of $X_{\omega}$ , respectively. In particular, $\dim\nu_{\omega}$ is positive and constant for $\mu$ -almost every $\omega\in\Omega_{1}$ .

Corollary 5.10.

Let $\mu$ be an ergodic $\Gamma$ -invariant percolation with indistinguishable infinite clusters. Assume one of the following conditions:

•

$\mu$ is a supercritical Bernoulli percolation.
•

$\mu$ -almost every $\omega$ has a unique infinite cluster.
•

$\mu$ -almost every $\omega$ has a cluster with at least three ends.
•

The expected degree at $1$ on $\Omega_{1}$ is larger than $d-i(G)$ , where $i(G)$ is the isoperimetric constant of $G$ .

Then, for $\mu$ -almost every $\omega\in\Omega_{1}$ , letting $\nu_{\omega}$ be the harmonic measure on $\partial G$ determined by $X_{\omega}$ , we have

\dim\nu_{\omega}=\lim_{r\to 0}\frac{\log\nu_{\omega}(B(\xi,r))}{\log r}=\frac{h}{l}

for $\nu_{\omega}$ -almost every $\xi\in\partial G$ . In particular, $\dim\nu_{\omega}$ is positive and constant for $\mu$ -almost every $\omega\in\Omega_{1}$ .

Proof.

This follows from Theorem 5.6 and Theorem 5.9. ∎

In the rest of this section, we prove Theorem 5.9. First, we have the following results as in the case of random conductance models.

Theorem 5.11.

Let $\mu$ be an ergodic $\Gamma$ -invariant percolation on $G$ with indistinguishable infinite clusters. Assume that the corresponding simple random walk has positive speed. Then, for $\mu$ -almost every $\omega\in\Omega_{1}$ and for $\nu_{\omega}$ -almost every $\xi\in\partial G$ , we have

\limsup_{r\to 0}\frac{\log\nu_{\omega}(B(\xi,r))}{\log r}\leq\frac{h}{l}

where $h$ and $l$ are the entropy and the speed of $X_{\omega}$ , respectively. In particular, we have $\dim\nu_{\omega}\leq\frac{h}{l}$ for $\mu$ -almost every $\omega\in\Omega_{1}$ .

Proof.

This follows from Theorem 3.1 and the assumption that the speed is positive. ∎

Theorem 5.12.

Let $\mu$ be an ergodic $\Gamma$ -invariant percolation on $G$ with indistinguishable infinite clusters. Assume that the simple random walk on $C_{\omega}(1)$ has positive speed. Then, for every $\epsilon>0$ , the subset $D_{\epsilon}$ of $\Xi$ defined by

D_{\epsilon}:=\biggl{\{}(\omega,\xi)\in\Omega_{1}\times\partial G\colon\liminf_{r\to 0}\frac{\log\nu_{\omega}(B(\xi,r))}{\log r}\geq\frac{h}{l}-\epsilon\biggr{\}}.

is $\eta$ -positive, where $\eta=\int_{\Omega_{1}}\delta_{\omega}\otimes\nu_{\omega}d\mu^{\prime}(\omega)$ is the probability measure defined in Definition 5.7.

Proof.

This follows from Theorem 3.2 and the assumption that the speed is positive. ∎

In the proof of Theorem 3.1 in [Tan19], the ergodicity of the $\Gamma$ -action on $\partial G$ is used to prove that $D_{\epsilon}$ is actually full measure. We can modify the argument by using ergodicity of the equivalence relation $\mathcal{R}_{1}$ defined in Lemma 5.8.

Proof of Theorem 5.9.

Let us show that the set $D_{\epsilon}$ in Theorem 5.12 is $\eta$ -conull for every $\epsilon>0$ . It is enough to prove that $D_{\epsilon}$ is $\mathcal{R}_{1}$ -invariant since $\mathcal{R}_{1}$ is ergodic with respect to $\eta$ (Lemma 5.8) and $D_{\epsilon}$ is $\eta$ -positive (Theorem 5.12). Let $((\omega,\xi),(g\omega,g\xi))$ be an element in $\mathcal{R}_{1}|_{\Omega_{1}}$ and $(\omega,\xi)\in D_{\epsilon}$ . For $g\in\Gamma$ , let $\mathbb{P}_{\omega,g}$ denotes the distribution of sample paths starting from $g$ following $P_{\omega}$ . Then, for every Borel subset $A\subset\partial G$ , we have

	$\displaystyle(g_{*}\nu_{\omega})(A)$	$\displaystyle=\mathbb{P}_{\omega}(\{x\in\Gamma^{{\mathbb{Z}}_{+}}\colon\xi_{x}\in g^{-1}A\})\geq\mathbb{P}_{\omega}(\bigcup_{n\in{\mathbb{Z}}_{+}}\{x_{n}=g^{-1}\}\cap\{x\in\Gamma^{{\mathbb{Z}}_{+}}\colon\xi_{x}\in g^{-1}A\})$
		$\displaystyle=\mathbb{P}_{\omega}(\bigcup_{n\in{\mathbb{Z}}_{+}}\{x_{n}=g^{-1}\})\mathbb{P}_{\omega,g^{-1}}(\{x\in\Gamma^{{\mathbb{Z}}_{+}}\colon\xi_{x}\in g^{-1}A\})$
		$\displaystyle=\mathbb{P}_{\omega}(\bigcup_{n\in{\mathbb{Z}}_{+}}\{x_{n}=g^{-1}\})\mathbb{P}_{g\omega}(\{x\in\Gamma^{{\mathbb{Z}}_{+}}\colon\xi_{x}\in A\})=\mathbb{P}_{\omega}(\bigcup_{n\in{\mathbb{Z}}_{+}}\{x_{n}=g^{-1}\})\nu_{g\omega}(A),$

where we have used the strong Markov property of $X_{\omega}$ in the second equality. The third equality follows from the equality $\mathbb{P}_{g\omega}=g_{*}\mathbb{P}_{\omega,g^{-1}}$ . Note that $\mathbb{P}_{\omega}(\bigcup_{n\in{\mathbb{Z}}_{+}}\{x_{n}=g^{-1}\})>0$ since $(\omega,g\omega)\in\mathcal{R}^{cl}$ . Further, by Lemma 2.4, we have $\nu_{g\omega}(B(g\xi,r))\leq\nu_{g\omega}(gB(\xi,c_{g}r))$ where $c_{g}$ is a constant depending only on $g$ . Then, we have

	$\displaystyle\liminf_{r\to 0}\frac{\log\nu_{g\omega}(B(g\xi,r))}{\log r}$	$\displaystyle\geq\liminf_{r\to 0}\frac{\log(g_{*}\nu_{\omega})(gB(\xi,c_{g}r))-\log\mathbb{P}_{\omega}(\bigcup_{n\in{\mathbb{Z}}_{+}}\{x_{n}=g^{-1}\})}{\log r}$
		$\displaystyle=\liminf_{r\to 0}\frac{\log\nu_{\omega}(B(\xi,c_{g}r))}{\log r}=\liminf_{r\to 0}\frac{\log\nu_{\omega}(B(\xi,r))}{\log r}$
		$\displaystyle\geq\frac{h}{l}-\epsilon$

from the above estimates and $(\omega,\xi)\in D_{\epsilon}$ . Hence, it follows that $(g\omega,g\xi)\in D_{\epsilon}$ , as required. ∎

6. Application to hyperbolic tilings

Let $\mathbb{H}^{2}$ be the hyperbolic plane and $o$ be its base point. For the pair of positive integers $P$ and $Q$ with $1/P+1/Q<1/2$ , let $T_{P,Q}$ be the regular tiling of $\mathbb{H}^{2}$ by $P$ -gons with interior angles $2\pi/Q$ , and $(\Gamma_{P,Q},S_{P,Q})$ be the pair of the cocompact Fuchsian group and its generating set corresponding to $T_{P,Q}$ . Let $G_{P,Q}$ be the Cayley graph of $(\Gamma_{P,Q},S_{P,Q})$ . In this section, we always consider the metric $d_{\mathbb{H}}$ on $G_{P,Q}$ induced from the standard hyperbolic metric on $\mathbb{H}^{2}$ by identifying $g$ with $go$ for $g\in\Gamma_{P,Q}$ .
In [CLP21], Carrasco, Lessa, and Paquette gave a lower bound for the speed of the simple random walks in Bernoulli percolation clusters on $G_{P,Q}$ , with respect to $d_{\mathbb{H}}$ .

Theorem 6.1 (Theorem 3.3 in [CLP21]).

Let $\mu_{p,P,Q}$ be a supercritical Bernoulli percolation on $G_{P,Q}$ with $p\in[0,1]$ , and $l_{p,P,Q}$ be the speed of the simple random walks determined by $\mu_{p,P,Q}$ , with respect to $d_{\mathbb{H}}$ . Then we have

l_{p,P,Q}\geq 2\log Q-\frac{1}{p}O(\log(\log Q))

when $Q\to\infty$ , uniformly in $P$ .

Combining this theorem with our result, we obtain the dimension drop on $G_{P,Q}$ for all large $Q$ .

Theorem 6.2.

Let $\mu_{p,P,Q}$ be a supercritical Bernoulli percolation on $G_{P,Q}$ with $p\in[0,1]$ . Then, for $\mu_{p,P,Q}$ -almost every $\omega\in\Omega_{1}$ , letting $\nu_{\omega}$ be the harmonic measure associated with the simple random walk on $C_{\omega}(1)$ starting from 1, we have

\dim\nu_{\omega}=\lim_{r\to 0}\frac{\log\nu_{\omega}(B(\xi,r))}{\log r}=\frac{h}{l}

for $\nu_{\omega}$ -almost every $\xi$ . Further we have

\limsup_{Q\to\infty}\delta_{p,P,Q}\leq\frac{1}{2}

uniformly in $P$ , where $\delta_{p,P,Q}$ denotes the dimension of $\nu_{\omega}$ for $\nu_{p,P,Q}$ -almost every $\omega\in\Omega_{1}$ .

Proof.

The first part of the theorem can be shown by the same argument as in the proof of Theorem 5.9 (just replacing the word metric with $d_{\mathbb{H}}$ ). Further, we obtain the inequality from Theorem 6.1 and the trivial bound $h_{p,P,Q}\leq\log Q$ . ∎

7. Questions and Remarks

In this section, we propose some natural questions concerning our results and related works. First, we review dimension drop phenomena. In various settings, it has been shown that the dimension of the harmonic measure associated with a random walk is strictly smaller than the dimension of the boundary (see [LPP95], [Led01], [GMM18], [Kos21], [CLP21], and [KT22]). Such a phenomenon is called dimension drop. In the context of random walks in percolation clusters, we can consider the stronger version of dimension drop phenomena.

Question 7.1.

Let $\mu$ a $\Gamma$ -invariant percolation on $G$ as in Theorem 5.9. When is the inequality

\dim\nu_{\omega}\leq\dim_{\rho}\Lambda(C_{\omega}(1))

strict for $\mu$ -almost every $\omega$ ? Here $\Lambda(C_{\omega}(1))$ denotes the limit set of $C_{\omega}(1)$ , i.e., $\overline{C_{\omega}(1)}^{G\cup\partial G}\backslash G$ . We can also consider the analogous question in the setting of hyperbolic tilings.

Note that Theorem 6.2 can be seen as a partial answer to this question in the case of Bernoulli percolations on hyperbolic tilings.

Next, we focus on Bernoulli percolations. In [Lal01], Lalley investigated the behavior of the dimension of the limit sets of percolation clusters of $\mu_{p}$ when the parameter $p$ varies, and obtained the monotonicity and the continuity. We can consider the following analogue of his results:

Question 7.2.

Let $\mu_{p}$ be a supercritical Bernoulli percolation on $G$ and $\delta_{p}$ be the dimension of harmonic measures associated with $\mu_{p}$ . Is the function $p\mapsto\delta_{p}$ continuous or monotone increasing?

By Theorem 5.9, the continuity of the dimension can be reduced to that of the entropy and the speed. I do not know anything about the monotonicity of them. Note that the monotonicity of the entropy was conjectured in [BLS99] and it still remains open. See also a recent paper by Lyons and White [LW23].

Appendix A Proof of Theorem 5.5

In this appendix, we give the proof of Theorem 5.5. Similar arguments work for Theorem 4.3.
Before the proof, we first define the bilateral version $(\Pi_{{\mathbb{Z}}},\lambda_{{\mathbb{Z}}},U)$ of the system $(\Pi,\lambda,T)$ by

•

$\Pi_{{\mathbb{Z}}}=\Omega_{1}\times\Gamma^{{\mathbb{Z}}}$
•

$\lambda_{{\mathbb{Z}}}=\int_{\Omega_{1}}\delta_{\omega}\otimes\mathbb{P}^{{\mathbb{Z}}}_{\omega}d\mu^{\prime}(\omega)$
•

$U(\omega,(x_{n})_{n\in{\mathbb{Z}}})=(x_{1}^{-1}\omega,(x_{1}^{-1}x_{n+1})_{n\in{\mathbb{Z}}})$ ,

where $\mathbb{P}^{{\mathbb{Z}}}_{\omega}$ denotes the distribution of bilateral sample paths following $P_{\omega}$ starting from 1.

Proof of Theorem 5.5.

It is enough to show that the bilateral version is measure-preserving and ergodic since the natural projection from the bilateral version to $(\Pi,\lambda,T)$ is a factor map commuting with the shift maps.
First, we show that the bilateral system is measure-preserving. Let $B$ be a Borel subset of $\Pi$ . Then, we have

	$\displaystyle\lambda(U^{-1}B)$	$\displaystyle=\frac{1}{D}\int_{\Omega_{1}}\deg_{C_{\omega}(1)}(1)\mathbb{P}^{{\mathbb{Z}}}_{\omega}(\{x\in\Gamma^{{\mathbb{Z}}}\colon(\omega,x)\in U^{-1}B\}d\mu(\omega)$
		$\displaystyle=\frac{1}{D}\sum_{s\in S}\int_{\Omega_{1}}\omega(1,s)\mathbb{P}^{{\mathbb{Z}}}_{\omega}(\{x\in\Gamma^{{\mathbb{Z}}}\colon(s^{-1}\omega,(s^{-1}x_{n+1})_{n\in{\mathbb{Z}}})\in B\}\|\{x_{1}=s\})d\mu(\omega)$
		$\displaystyle=\frac{1}{D}\sum_{s\in S}\int_{\Omega_{1}}(s^{-1}\omega)(1,s^{-1})\mathbb{P}^{{\mathbb{Z}}}_{s^{-1}\omega}(\{x\in\Gamma^{{\mathbb{Z}}}\colon(s^{-1}\omega,x)\in B\}\|\{x_{-1}=s^{-1}\})d\mu(\omega)$
		$\displaystyle=\frac{1}{D}\sum_{s\in S}\int_{\Omega_{1}}\omega(1,s^{-1})\mathbb{P}^{{\mathbb{Z}}}_{\omega}(\{x\in\Gamma^{{\mathbb{Z}}}\colon(\omega,x)\in B\}\|\{x_{-1}=s^{-1}\})d\mu(\omega)$
		$\displaystyle=\frac{1}{D}\int_{\Omega_{1}}\deg_{C_{\omega}(1)}(1)\mathbb{P}^{{\mathbb{Z}}}_{\omega}(\{x\in\Gamma^{{\mathbb{Z}}}\colon(\omega,x)\in B\})d\mu(\omega)=\lambda(B),$

where the fourth equality follows from the $\Gamma$ -invariance of $\mu$ and the fifth equality follows from $S=S^{-1}$ . This completes the proof that $U$ is measure-preserving.
We prove that the bilateral system is ergodic in the rest of the proof, following Section 2 in [BB07]. For each Borel subset $B\subset\Pi_{{\mathbb{Z}}}$ , we define a function $f_{B}$ on $\Omega_{1}$ by $f_{B}(\omega)=\mathbb{P}^{{\mathbb{Z}}}_{\omega}(\{x\in\Gamma^{{\mathbb{Z}}}\colon(\omega,x)\in B\})$ . Let $A$ be a $U$ -invariant subset of $\Pi_{{\mathbb{Z}}}$ . We first show that

f_{A}(\omega)\in\{0,1\}

for $\mu^{\prime}$ -almost every $\omega\in\Omega_{1}$ . For $\epsilon>0$ , there exist $N\in{\mathbb{N}}$ and a Borel subset $A_{N}$ of $\Pi_{{\mathbb{Z}}}$ such that

•

$\lVert f_{A}-f_{A_{N}}\rVert_{1}$ = $\lambda_{{\mathbb{Z}}}(A\,\triangle\,A_{N})<\epsilon$ , and
•

$A_{N}$ is an event depending only on $(\omega,(x_{n})_{-N\leq n\leq N})$ ,

where $\lVert\cdot\rVert_{1}$ denotes the norm on $L^{1}_{\mu^{\prime}}(\Omega_{1})$ . Since $A$ is $U$ -invariant, we have

	$\displaystyle\lVert f_{A}-f_{A}^{2}\rVert_{1}$	$\displaystyle=\lVert f_{A}-f_{U^{N}A}f_{U^{-N}A}\rVert_{1}\leq\lVert f_{A}-f_{U^{N}A_{N}}f_{U^{-N}A_{N}}\rVert_{1}+2\epsilon$
		$\displaystyle\leq\lVert f_{A}-f_{U^{N}A_{N}\cap U^{-N}A_{N}}\rVert_{1}+2\epsilon$
		$\displaystyle\leq\lVert f_{A}-f_{U^{N}A\cap U^{-N}A}\rVert_{1}+4\epsilon=\lVert f_{A}-f_{A}\rVert_{1}+4\epsilon=4\epsilon.$

Note that the second inequality follows from the fact that the events $U^{N}A_{N}$ and $U^{-N}A_{N}$ are independent, conditional on $\omega$ . Since $\epsilon$ can be taken arbitrarily, the above estimate implies that $f_{A}(\omega)\in\{0,1\}$ for $\mu^{\prime}$ -almost every $\omega\in\Omega_{1}$ . Therefore, it is enough to show that the subset $F$ of $\Omega_{1}$ defined by

F=\{\omega\in\Omega_{1}\colon f_{A}(\omega)=1\}

is $\mathcal{R}^{cl}$ -invariant since $\mathcal{R}^{cl}|_{\Omega_{1}}$ is ergodic with respect to $\mu^{\prime}$ . We want to show that $s^{-1}\omega\in F$ for $\omega\in F$ and $s\in S$ satisfying $\omega(1,s)=1$ . Since $\omega\in F$ , we have

	$\displaystyle f_{A}(s^{-1}\omega)$	$\displaystyle=\mathbb{P}^{{\mathbb{Z}}}_{s^{-1}\omega}(\{x\in\Gamma^{{\mathbb{Z}}}\colon(s^{-1}\omega,x)\in A\})=s^{-1}_{*}\mathbb{P}^{{\mathbb{Z}}}_{\omega,s}(\{x\in\Gamma^{{\mathbb{Z}}}\colon(s^{-1}\omega,x)\in A\})$
		$\displaystyle=\mathbb{P}^{{\mathbb{Z}}}_{\omega,s}(\{x\in\Gamma^{{\mathbb{Z}}}\colon(s^{-1}\omega,s^{-1}x)\in A\})\geq\mathbb{P}^{{\mathbb{Z}}}_{\omega}(\{x\in\Gamma^{{\mathbb{Z}}}\colon(\omega,x)\in A,\,x_{1}=s\})$
		$\displaystyle=\mathbb{P}^{{\mathbb{Z}}}_{\omega}(\{x\in\Gamma^{{\mathbb{Z}}}\colon x_{1}=s\})>0,$

where $\mathbb{P}^{{\mathbb{Z}}}_{\omega,s}$ denotes the distribution of bilateral sample paths following $P_{\omega}$ starting from $s$ . Note that we have used the $U$ -invariance of $A$ and the strong Markov property of $X_{\omega}$ in the first inequality. As $f_{A}(\omega)\in\{0,1\}$ , this implies $f_{A}(s^{-1}\omega)=1$ , and hence completes the proof. ∎

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Dimensions of harmonic measures in percolation clusters on hyperbolic groups

Abstract.

2020 Mathematics Subject Classification:

1. Introduction

1.1. Background and main results

Theorem 1.1.

Theorem 1.2.

1.2. Outline of the proof

1.3. Related works and applications

Theorem 1.3.

1.4. Organization of the paper

Acknowledgements

2. Preliminaries

2.1. Geometry of word hyperbolic groups

Definition 2.1 (δ\delta-hyperbolicity).

Definition 2.2 (Hyperbolic groups).

Definition 2.3 (Gromov boundary).

Lemma 2.4 (Lemma 2.2 in [Tan19]).

Definition 2.5.

Lemma 2.6 (Proposition 2.1 in [BHM11]).

2.2. Hausdorff dimensions of measures

Definition 2.7 (Hausdorff dimensions of measures).

Lemma 2.8.

Proof.

Definition 2.9 (Exact dimensionality).

2.3. Regular Markov chains on hyperbolic groups

Definition 2.10 (Regular Markov chains).

Theorem 2.11 (Section 7 of [Kai00]).

3. General Estimates for Dimensions

Theorem 3.1.

Theorem 3.2.

Proof of Theorem 3.1..

Proof of Theorem 3.2..

4. Invariant Random Conductance Models

4.1. Ergodic Theory of Random Conductance Models

Definition 4.1 (Γ\Gamma-invariant random conductance models).

Definition 4.2 (Path bundles).

Theorem 4.3.

Theorem 4.4.

Proof.

Definition 4.5 (Ray bundles).

Lemma 4.6.

Proof.

4.2. The Dimension Formula for Invariant Random Conductance Models

Theorem 4.7.

Proof.

Theorem 4.8.

Proof.

Theorem 4.9.

Proof.

5. Invariant Percolations

5.1. Ergodic Theory of Invariant Percolations

Definition 5.1 (Γ\Gamma-invariant percolations).

Definition 5.2 (Cluster relations and indistinguishability).

Remark 5.3.

Definition 5.4 (Path bundles).

Theorem 5.5.

Theorem 5.6 (Theorem 4.4 in [BLS99]).

Definition 5.7 (Ray bundles).

Lemma 5.8.

Proof.

5.2. The Dimension Formula for Invariant Percolations

Theorem 5.9.

Corollary 5.10.

Proof.

Theorem 5.11.

Proof.

Theorem 5.12.

Proof.

Proof of Theorem 5.9.

6. Application to hyperbolic tilings

Theorem 6.1 (Theorem 3.3 in [CLP21]).

Theorem 6.2.

Proof.

7. Questions and Remarks

Question 7.1.

Question 7.2.

Appendix A Proof of Theorem 5.5

Proof of Theorem 5.5.

References

Definition 2.1 ( $\delta$ -hyperbolicity).

Definition 4.1 ( $\Gamma$ -invariant random conductance models).

Definition 5.1 ( $\Gamma$ -invariant percolations).