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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.

Dimensions of harmonic measures in percolation clusters on hyperbolic groups

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.
Paper Structure (19 sections, 24 theorems, 73 equations)

This paper contains 19 sections, 24 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Background and main results
  3. Outline of the proof
  4. Related works and applications
  5. Organization of the paper
  6. Preliminaries
  7. Geometry of word hyperbolic groups
  8. Hausdorff dimensions of measures
  9. Regular Markov chains on hyperbolic groups
  10. General Estimates for Dimensions
  11. Invariant Random Conductance Models
  12. Ergodic Theory of Random Conductance Models
  13. The Dimension Formula for Invariant Random Conductance Models
  14. Invariant Percolations
  15. Ergodic Theory of Invariant Percolations
  16. ...and 4 more sections

Key Result

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

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1: $\delta$-hyperbolicity
  • Definition 2.2: Hyperbolic groups
  • Definition 2.3: Gromov boundary
  • Lemma 2.4: Lemma 2.2 in MR3893268
  • Definition 2.5
  • Lemma 2.6: Proposition 2.1 in MR2919980
  • Definition 2.7: Hausdorff dimensions of measures
  • ...and 44 more