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

Abstract

For every 3/4 ≤ δ, β < 1 satisfying δ ≤ β < 1+δ 2 we construct a finitely generated group Γ and a (symmetric, finitely supported) random walk X n on Γ so that its expected distance from its starting point satisfies E|X n | ≍ n β and its entropy satisfies H(X n ) ≍ n δ . In fact, the speed and entropy can be set precisely to equal any two nice enough prescribed functions f, h up to a constant factor as long as the functions satisfy the relation n 3 4 ≤ h(n) ≤ f (n) ≤ nh(n)/log(n + 1) ≤ n γ for some γ < 1.

Isoperimetric profiles and random walks on some groups defined by piecewise actions

“…Some recent results on harmonic functions and group structure are in [2,4,5,7,19,21,22,26]. Entropy and/or displacement are discussed in [1,6,12,16,27,29]. Probability of return, spectral and other properties are discussed in [3,8,9,11,20,[35][36][37].…”

Isoperimetric profiles and random walks on some groups defined by piecewise actions

“…Some recent results on harmonic functions and group structure are in [2,19,7,26,4,22,5,21]. Entropy and/or displacement are discussed in [16,6,1,30,28,12]. Probability of return, spectral and other properties are discussed in [9,11,3,8,36,37,38,20].…”

The Liouville property and Hilbertian compression

“…These constructions often leave, in the meantime, the other parameters uncomputed (and hard to compute). It might, for this precise reason be even more interesting to have bounds between those quantities (see Amir [1] or Brieussel & Zheng [12] for recent developments). In fact, too good improvements of the bounds in Theorem 1.3 would lead to some forms of the gap conjecture on volume growth.…”