Non-Liouville groups with return probability exponent at most 1/2

Michał Kotowski,Bálint Virág

doi:10.1214/ecp.v20-3774

Michał Kotowski, Bálint Virág

Open Access

https://doi.org/10.1214/ecp.v20-3774

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Abstract

We construct a finitely generated group $G$ without the Liouville property such that the return probability of a random walk satisfies $p_{2n}(e,e) \gtrsim e^{-n^{1/2+ o(1)}}$. This shows that the constant $1/2$ in a recent theorem by Saloff-Coste and Zheng, saying that return probability exponent less than $1/2$ implies the Liouville property, cannot be improved. Our construction is based on permutational wreath products over tree-like Schreier graphs and the analysis of large deviations of inverted orbits on such graphs.

Highlights

One of the basic topics of study in probability and group theory is the behavior of random walks on Cayley graphs of finitely generated groups
The average distance Ed(X0, Xn) of the random walk from the origin after n steps may grow linearly with n, in which case we say that the random walk has positive speed, or slower, in which case we say that the random walk has zero speed
Note that the exponents γ and β as above need not exist, so in general one should speak about lim inf and lim sup exponents

Summary

Introduction

One of the basic topics of study in probability and group theory is the behavior of random walks on Cayley graphs of finitely generated groups. A classical result (see for example discussion in [Pet, Chapter 9]) says that for groups (though not for general transitive graphs) having positive speed is equivalent to non-Liouville property Note, that it is not known if this property is independent of the generating set (or, more generally, the step distribution of the random walk), which is in contrast to return probabilities, whose decay rate is stable under quasi-isometries ([PSC00]). In general it would be desirable to obtain a better understanding of inverted orbits and probabilistic parameters (return probabilities, speed, entropy) on related groups of this type Some results along these lines can be found for example in [Bri13], where entropy and return probability exponents on groups of directed automorphisms of bounded degree trees are analyzed. We will use the notation f (n) g(n) meaning f (n) ≤ Cg(n) for some constant C > 0

Preliminaries

The bubble group

Bounds on the inverted orbits

Liouville property and transience

Findings

Lower bound on return probability