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Phase transition for recurrence of stationary random walks on lamplighter groups

Itai Benjamini, Guy Blachar, Ariel Yadin

TL;DR

This paper analyzes stationary random walks on lamplighter groups $H\wr G$ with switch-walk-switch lamp updates and a base drift on $G$. Using a decomposition into base excursions and an electrical-network perspective, it establishes a recurrence–transience phase transition when the base is virtually $\mathbb{Z}$ and the lamp group is finite, with a precise threshold depending on $|H|$ (and in the $G=\mathbb{Z}$ case, an exact critical parameter). It further shows that when $G$ is not virtually $\mathbb{Z}$ or when $H$ is infinite, the walks are transient for all drift parameters, and extends the analysis to general lamplighter graphs via homesick random walks, including sharp local-time asymptotics on $\mathbb{Z}$ and concentration results for the range. The results highlight how base-graph growth controls the lamplighter dynamics and demonstrate that recurrence phenomena arise only in effectively one-dimensional settings, with robust independence from lamp-measure details in the finite-lamp case. These findings contribute to the broader understanding of lamplighter dynamics, phase transitions, and the role of drift in random walks on groups and graphs.

Abstract

We introduce and study a class of random walks on lamplighter groups $H\wr G$, where $H$ is a nontrivial finitely generated group and $G$ is an infinite finitely generated group, called \textbf{stationary random walks}. At each step, the walk switches the lamp at its current position, moves in the base group with a drift towards the identity, and switches the lamp again at the new position. We show that when $G$ is virtually-$\mathbb{Z}$ and $H$ is finite, these walks exhibit a phase transition between recurrence and transience, while when~$G$ is not virtually-$\mathbb{Z}$ or $H$ is infinite, they are always transient. In the case $G=\mathbb{Z}$, we determine the exact critical parameter and provide a quantitative description of this phase transition.

Phase transition for recurrence of stationary random walks on lamplighter groups

TL;DR

This paper analyzes stationary random walks on lamplighter groups with switch-walk-switch lamp updates and a base drift on . Using a decomposition into base excursions and an electrical-network perspective, it establishes a recurrence–transience phase transition when the base is virtually and the lamp group is finite, with a precise threshold depending on (and in the case, an exact critical parameter). It further shows that when is not virtually or when is infinite, the walks are transient for all drift parameters, and extends the analysis to general lamplighter graphs via homesick random walks, including sharp local-time asymptotics on and concentration results for the range. The results highlight how base-graph growth controls the lamplighter dynamics and demonstrate that recurrence phenomena arise only in effectively one-dimensional settings, with robust independence from lamp-measure details in the finite-lamp case. These findings contribute to the broader understanding of lamplighter dynamics, phase transitions, and the role of drift in random walks on groups and graphs.

Abstract

We introduce and study a class of random walks on lamplighter groups , where is a nontrivial finitely generated group and is an infinite finitely generated group, called \textbf{stationary random walks}. At each step, the walk switches the lamp at its current position, moves in the base group with a drift towards the identity, and switches the lamp again at the new position. We show that when is virtually- and is finite, these walks exhibit a phase transition between recurrence and transience, while when~ is not virtually- or is infinite, they are always transient. In the case , we determine the exact critical parameter and provide a quantitative description of this phase transition.

Paper Structure

This paper contains 7 sections, 16 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Recurrence and transience over Z
  3. Local time over Z
  4. Homesick random walks
  5. Stationary random walks: the general case
  6. Proofs of Theorem \ref{['thm:virt-Z']} and Theorem \ref{['thm:no-transition']}
  7. Stationary random walks on lamplighter graphs

Key Result

Theorem 1.1

Let $R_n$ denote the stationary random walk on $F\wr\mathbb{Z}$ with parameter $p$. Then $R_n$ is recurrent if $p\ge\frac{\left|F\right|^2}{\left|F\right|^2+1}$, and is transient if $p<\frac{\left|F\right|^2}{\left|F\right|^2+1}$.

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 2.1
  • Proposition 2.2
  • proof
  • ...and 29 more