Non-Liouville groups with return probability exponent at most 1/2. (English) Zbl 1318.60007

Kotowski, Michał; Virág, Bálint

Electron. Commun. Probab. 20, Paper No. 12, 12 p. (2015).

Summary: 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 L. Saloff-Coste and T. Zheng [“Random walks and isoperimetric profiles under moment conditions”, Preprint, arXiv:1501.05929], 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.

MSC:

60B15	Probability measures on groups or semigroups, Fourier transforms, factorization
20F65	Geometric group theory

Keywords:

permutational wreath products; random walks; return probabilities