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 $p_{2n}(e,e)\gtrsim e^{-n^{1/2+o(1)}}$. 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.

15 pages, 1 figure; v2: minor corrections

References in corpus (2)

• Behaviors of entropy on finitely generated groups
• Imbeddings into groups of intermediate growth