The Liouville property and Hilbertian compression

Abstract

Lower bound on the equivariant Hilbertian compression exponent α are obtained using random walks. More precisely, if the probability of return of the simple random walk is expThis motivates the study of further relations between return probability, speed, entropy and volume growth. For example, ifUnder a strong assumption on the off-diagonal decay of the heat kernel, the lower bound on compression improves to α ≥ 1 − γ. Using a result from Naor & Peres [27] on compression and the speed of random walks, this yields very promising bounds on speed and implies the Liouville property if γ < 1/2.

On Groups, Slow Heat Kernel Decay Yields Liouville Property and Sharp Entropy Bounds

“…The idea of connecting slow decay of the return probability to the Liouville property was first proposed by Gournay in [Gou16]. The Hilbert compression exponent α * 2 (G) was introduced in Guentner and Kaminker [GK04], who defined α * 2 (G) as the supremum over all α ≥ 0 such that there exists a 1-Lipschitz map f : G → H and a constant c > 0 such that f (x) − f (y) H ≥ cd G (x, y) α .…”

“…It is known that for finitely generated amenable groups, α * 2 (G) = α # 2 (G), see [dCTV07]. One result from [Gou16] is the following. Suppose µ is a symmetric probability measure of finite support on G such that…”

On the joint behaviour of speed and entropy of random walks on groups

“…the references in section 1), but very few constructions allowing to control more than one of them simultaniously. For more on the various exponents and their relations we refer the reader to [11].…”

Random walks and isoperimetric profiles under moment conditions

“…Remark 2.12. Theorem 2.9 is related to [13, Theorem 1.4] and [13, Conjecture 1.5] and some of the ingredients of the proof given below are similar to those used in [13]. The hypothesis (OD) appearing in [13, Theorem 1.4] plays no role in Theorem 2.9.…”