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

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.