NonLiouville 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 nonLiouville groups. Our construction is based on permutational wreath products over treelike Schreier graphs and the analysis of large deviations of inverted orbits on such graphs.
 Publication:

arXiv eprints
 Pub Date:
 August 2014
 arXiv:
 arXiv:1408.6895
 Bibcode:
 2014arXiv1408.6895K
 Keywords:

 Mathematics  Group Theory;
 Mathematics  Probability
 EPrint:
 15 pages, 1 figure