Law of large numbers for the drift of two-dimensional wreath product

TL;DR

This paper establishes a law of large numbers for the drift of random walks on the two-dimensional lamplighter group, highlighting differences with classical abelian groups and exploring the relation with group geometry.

Contribution

It proves the LLN for the drift in the lamplighter group under finite moment conditions and compares this with other groups to understand geometric influences.

Findings

LLN holds for the drift in the lamplighter group with finite $(2+\epsilon)$-moment.

Displacement distribution differs from classical abelian groups, showing concentration.

Explores the connection between LLN properties and the asymptotic geometry of groups.

Abstract

We prove the law of large numbers for the drift of random walks on the two-dimensional lamplighter group, under the assumption that the random walk has finite $(2+\epsilon)$-moment. This result is in contrast with classical examples of abelian groups, where the displacement after $n$ steps, normalised by its mean, does not concentrate, and the limiting distribution of the normalised $n$-step displacement admits a density whose support is $[0,\infty)$. We study further examples of groups, some with random walks satisfying LLN for drift and other examples where such concentration phenomenon does not hold, and study relation of this property with asymptotic geometry of groups.