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

TL;DR

This paper constructs a finitely generated non-Liouville group with a return probability decay exponent close to 1/2, providing insights into the minimal possible decay rate for such groups.

Contribution

It introduces a new construction of non-Liouville groups with near-optimal return probability decay, advancing understanding of their probabilistic properties.

Findings

Constructed a non-Liouville group with return probability exponent near 1/2

Analyzed large deviations of inverted orbits on Schreier graphs

Provided evidence that 1/2 is the smallest possible exponent for non-Liouville groups

Abstract

We construct a finitely generated group $G$ without the Liouville property such that the return probability of a random walk satisfies $p_{2n}(e,e) \gtrsim e^{-n^{1/2 + o(1)}}$. Recent results suggest that $1/2$ is indeed the smallest possible return probability exponent for non-Liouville groups. 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.