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The Poisson boundary of wreath products

Joshua Frisch, Eduardo Silva

Abstract

We give a complete description of the Poisson boundary of wreath products $A\wr B= \bigoplus_{B} A\rtimes B$ of countable groups $A$ and $B$, for probability measures $μ$ with finite entropy where lamp configurations stabilize almost surely. If, in addition, the projection of $μ$ to $B$ is Liouville, we prove that the Poisson boundary of $(A\wr B,μ)$ is equal to the space of limit lamp configurations, endowed with the corresponding hitting measure. In particular, this answers an open question asked by Kaimanovich, and Lyons-Peres, for $B=\mathbb{Z}^d$, $d\ge 3$, and measures $μ$ with a finite first moment.

The Poisson boundary of wreath products

Abstract

We give a complete description of the Poisson boundary of wreath products of countable groups and , for probability measures with finite entropy where lamp configurations stabilize almost surely. If, in addition, the projection of to is Liouville, we prove that the Poisson boundary of is equal to the space of limit lamp configurations, endowed with the corresponding hitting measure. In particular, this answers an open question asked by Kaimanovich, and Lyons-Peres, for , , and measures with a finite first moment.
Paper Structure (30 sections, 34 theorems, 83 equations)

This paper contains 30 sections, 34 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Background
  4. Sketch of the proof of Theorems \ref{['thm: lamplighter Poisson boundary']} and \ref{['thm: Poisson for finite first moment']}
  5. Organization
  6. Preliminaries
  7. Wreath products and lamplighter groups
  8. Random walks and the Poisson boundary
  9. Entropy
  10. The conditional entropy criterion
  11. Properties of entropy
  12. Setup and the coarse trajectory
  13. Definition of the boundary
  14. Transience
  15. Main Assumptions
  16. ...and 15 more sections

Key Result

Theorem 1.3

Consider non-trivial countable groups $A$ and $B$. Let $\mu$ be a probability measure on $A\wr B$ with finite entropy and such that lamp configurations stabilize almost surely. Denote by $(\partial B,\nu_B)$ the Poisson boundary of the induced random walk on $B$. Then the Poisson boundary of $(A\wr

Theorems & Definitions (78)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Corollary 1.8
  • Corollary 1.9
  • Definition 2.1
  • Definition 2.2
  • ...and 68 more