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The Poisson boundary of lampshuffler groups

Eduardo Silva

TL;DR

This work analyzes random walks on the lampshuffler group ${\mathsf{Shuffler}}(H)=\mathrm{FSym}(H)\rtimes H$, focusing on the Poisson boundary. By proving a stabilization lemma under the assumption of finite first moment and transient projection to $H$, it shows that the permutation coordinate $F_n$ converges pointwise to a limit function $F_{\infty}$; for $H=\mathbb{Z}$, the Poisson boundary is completely described by the space of limit functions with the hitting measure. The paper further establishes boundary triviality for finitary measures when the base projection is recurrent on $\mathbb{Z}$ or $\mathbb{Z}^2$, and situates these results within the broader theory of wreath products and cyclic extensions, employing Kaimanovich's Conditional Entropy Criterion and a displacement analysis. Overall, it provides a new complete description of the Poisson boundary for a nontrivial class of locally finite-by-$\mathbb{Z}$ groups and highlights the delicate role of moment conditions and base-recurrence in boundary behavior.

Abstract

We study random walks on the lampshuffler group $\mathrm{FSym}(H)\rtimes H$, where $H$ is a finitely generated group and $\mathrm{FSym}(H)$ is the group of finitary permutations of $H$. We show that for any step distribution $μ$ with a finite first moment that induces a transient random walk on $H$, the permutation coordinate of the random walk almost surely stabilizes pointwise. Our main result states that for $H=\mathbb{Z}$, the above convergence completely describes the Poisson boundary of the random walk $(\mathrm{FSym}(\mathbb{Z})\rtimes \mathbb{Z},μ)$.

The Poisson boundary of lampshuffler groups

TL;DR

This work analyzes random walks on the lampshuffler group , focusing on the Poisson boundary. By proving a stabilization lemma under the assumption of finite first moment and transient projection to , it shows that the permutation coordinate converges pointwise to a limit function ; for , the Poisson boundary is completely described by the space of limit functions with the hitting measure. The paper further establishes boundary triviality for finitary measures when the base projection is recurrent on or , and situates these results within the broader theory of wreath products and cyclic extensions, employing Kaimanovich's Conditional Entropy Criterion and a displacement analysis. Overall, it provides a new complete description of the Poisson boundary for a nontrivial class of locally finite-by- groups and highlights the delicate role of moment conditions and base-recurrence in boundary behavior.

Abstract

We study random walks on the lampshuffler group , where is a finitely generated group and is the group of finitary permutations of . We show that for any step distribution with a finite first moment that induces a transient random walk on , the permutation coordinate of the random walk almost surely stabilizes pointwise. Our main result states that for , the above convergence completely describes the Poisson boundary of the random walk .
Paper Structure (17 sections, 16 theorems, 63 equations)

This paper contains 17 sections, 16 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Organization
  3. Acknowledgments
  4. Extensions of finitary symmetric groups
  5. Basic properties
  6. Wreath products subgroups in lampshufflers
  7. Cyclic extensions of locally finite groups
  8. Background on random walks and Poisson boundaries
  9. Random walks on groups
  10. The Poisson boundary
  11. The Poisson boundary of wreath products
  12. The conditional entropy criterion
  13. Random walks on ${\mathsf{Shuffler}}(H)$
  14. Stabilization of the permutation coordinate
  15. Proof of the main theorem
  16. ...and 2 more sections

Key Result

Lemma 1.1

Let $H$ be a finitely generated group, and consider a probability measure $\mu$ on ${\mathsf{Shuffler}}(H)$. Suppose that $\mu$ has a finite first moment and that it induces a transient random walk on $H$. Then for any $h\in H$, the values $F_n(h)$, $n\ge 0$, of the permutation coordinate of the ran

Theorems & Definitions (33)

  • Lemma 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 23 more