Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Entropies and Poisson boundaries of random walks on groups with rapid decay

Benjamin Anderson-Sackaney, Tim de Laat, Ebrahim Samei, Matthew Wiersma

Abstract

Let $G$ be a countable group and $μ$ a probability measure on $G$. We build a new framework to compute asymptotic quantities associated with the $μ$-random walk on $G$, using methods from harmonic analysis on groups and Banach space theory, most notably complex interpolation. It is shown that under mild conditions, the Lyapunov exponent of the $μ$-random walk with respect to a weight $ω$ on $G$ can be computed in terms of the asymptotic behavior of the spectral radius of $μ$ in an ascending class of weighted group algebras, and we prove that for natural choices of $ω$ and $μ$, the Lyapunov exponent vanishes. Also, we show that the Avez entropy of the $μ$-random walk can be realized as the Lyapunov exponent of $μ$ with respect to a suitable weight. We apply our results to stationary dynamical systems consisting of an action of a group with the property of rapid decay on a probability space. We prove that whenever the associated Koopman representation is weakly contained in the left-regular representation of the group, then the Avez entropy coincides with the Furstenberg entropy of the stationary space. This gives a characterization of (Zimmer) amenability for actions of rapid decay groups on stationary spaces. Next, by considering the spectral radius in the algebras of $p$-pseudofunctions on $G$, we introduce a new asymptotic quantity, which we call convolution entropy. We show that for groups with the property of rapid decay, the convolution entropy coincides with the Avez entropy.

Entropies and Poisson boundaries of random walks on groups with rapid decay

Abstract

Let be a countable group and a probability measure on . We build a new framework to compute asymptotic quantities associated with the -random walk on , using methods from harmonic analysis on groups and Banach space theory, most notably complex interpolation. It is shown that under mild conditions, the Lyapunov exponent of the -random walk with respect to a weight on can be computed in terms of the asymptotic behavior of the spectral radius of in an ascending class of weighted group algebras, and we prove that for natural choices of and , the Lyapunov exponent vanishes. Also, we show that the Avez entropy of the -random walk can be realized as the Lyapunov exponent of with respect to a suitable weight. We apply our results to stationary dynamical systems consisting of an action of a group with the property of rapid decay on a probability space. We prove that whenever the associated Koopman representation is weakly contained in the left-regular representation of the group, then the Avez entropy coincides with the Furstenberg entropy of the stationary space. This gives a characterization of (Zimmer) amenability for actions of rapid decay groups on stationary spaces. Next, by considering the spectral radius in the algebras of -pseudofunctions on , we introduce a new asymptotic quantity, which we call convolution entropy. We show that for groups with the property of rapid decay, the convolution entropy coincides with the Avez entropy.

Paper Structure

This paper contains 17 sections, 24 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Entropy of random walks
  4. Poisson boundary
  5. $L^p$-spaces and complex interpolation
  6. Weighted group algebras and $\ell^p$-spaces
  7. Groups with rapid decay
  8. Lyapunov exponent of random walks
  9. Vanishing of the Lyapunov exponent
  10. Weighted entropy of random walks
  11. Poisson boundaries and stationary spaces
  12. $L^p$-Koopman representation
  13. Stationary dynamical systems
  14. Convolution entropy
  15. Avez entropy as a convolution-type entropy
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $G$ be a countable group satisfying property RD with respect to a length function $\mathcal{L}$, and let $\mu$ be a non-degenerate probability measure on $G$ with both finite entropy and finite $\log (1+\mathcal{L})$-moment. Let $(X,\xi)$ be a $(G,\mu)$-stationary space such that the associated

Theorems & Definitions (60)

  • Theorem 1.1: Theorem \ref{['T:Square integrable-inverse Harich Chandra function']}
  • Theorem 1.2: Theorem \ref{['T:charc amen action-RD groups']}
  • Definition 1.3
  • Theorem 1.4: Theorem \ref{['T:Avez entropy-p conv operators-RD']}
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • ...and 50 more