For every satisfying we construct a finitely generated group and a (symmetric, finitely supported) random walk on so that its expected distance from its starting point satisfies and its entropy satisfies . In fact, the speed and entropy can be set precisely to equal any two nice enough prescribed functions up to a constant factor as long as the functions satisfy the relation for some .
Cite this article
Gideon Amir, On the joint behaviour of speed and entropy of random walks on groups. Groups Geom. Dyn. 11 (2017), no. 2, pp. 455–467DOI 10.4171/GGD/403