# Groups, Geometry, and Dynamics

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**Volume 11, Issue 2, 2017, pp. 455–467**

**DOI: 10.4171/GGD/403**

Published online: 2017-06-26

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

Gideon Amir^{[1]}(1) Bar-Ilan University, Ramat Gan, Israel

For every $3/4 \le \delta, \beta< 1$ satisfying $\delta\leq \beta < \frac{1+\delta}{2}$ we construct a finitely generated group $\Gamma$ and a (symmetric, finitely supported) random walk $X_n$ on $\Gamma$ so that its expected distance from its starting point satisfies $\mathbf E |X_n|\asymp n^{\beta}$ and its entropy satisfies $H(X_n)\asymp n^\delta$. 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 $n^{\frac{3}{4}}\leq h(n)\leq f(n)\leq \sqrt{{nh(n)}/{\log (n+1)}}\leq n^\gamma$ for some $\gamma<1$.

*Keywords: *Random walk, groups, entropy, rate of escape, permutation wreath product, automaton groups

Amir Gideon: On the joint behaviour of speed and entropy of random walks on groups. *Groups Geom. Dyn.* 11 (2017), 455-467. doi: 10.4171/GGD/403