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

Anna Erschler,Tianyi Zheng

doi:10.1007/s00440-021-01098-6

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.

Highlights

In this paper we study the limiting behavior of the distance to the origin of random walks on non-abelian solvable groups, in particular show a law of large numbers type result for random walks of sublinear drift on wreath products over Z2 with finite
We show a law of large numbers for the displacement of a centered random walk on the wreath product Z2 (Z/2Z)
We prove that the law of large numbers for random walk displacement holds in some examples with infinite lamp groups, for instance on the wreath product Z2 Z

Summary

Introduction

In this paper we study the limiting behavior of the distance to the origin of random walks on non-abelian solvable groups, in particular show a law of large numbers type result for random walks of sublinear drift on wreath products over Z2 with finite. See Proposition 6.6 for examples of finitely generated groups, on which a simple random walk (Xn)∞ n=0 satisfies that lS(Xn)/Lμ(n) converges in distribution to a limiting density with support (0, ∞) along one subsequence (ni ); and converges in probability to the constant 1 along another subsequence (mi ). Proposition 6.2 describes the limiting distribution of lS(Xn)/Lμ(n) for simple random walks on the lamplighter Z F over Z with finite lamp group F. As another evidence, in Lemma 6.4, we show that if G admits a sequence of controlled Følner pairs and limiting density of lS(Xn)/Lμ(n) exists, the support of the limiting density must be the whole ray (0, ∞). Given a finite set V ⊂ Z2, denote by ∂ V the inner boundary of V , that is, the set of points in V which have at least one neighbor site outside V

First observations about range of simple random walks on Z2

An auxiliary fact about TSP on the diluted lattices

E TS D2k

An uncrossing lemma

Wreath products with infinite lamp groups

Questions

The example of Z F

Dependence of the limiting distribution on subsequences