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

We construct a finitely generated group G without the Liouville property such that the return probability of a random walk satisfies p2n(e,e)≳e−n1/2+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. © 2015 University of Washington. All rights reserved.