Phase transition for recurrence of stationary random walks on lamplighter groups

2025/10/07 by Itaï Benjamini, Benjamini, Itai, Guy Blachar +3
Biochemistry, Genetics and Molecular Biology · Mathematics · #Diffusion and Search Dynamics #FOS: Mathematics #Group Theory (math.GR) #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2510.06441

openalex publication_date 2025/10/07 · openalex created_date 2025/10/13 · openalex updated_date 2026/07/28

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 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-ℤ and H is finite, these walks exhibit a phase transition between recurrence and transience, while when~G is not virtually-ℤ or H is infinite, they are always transient. In the case G=ℤ, we determine the exact critical parameter and provide a quantitative description of this phase transition.

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