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

Abstract

For every $3/4\le \delta ,\beta <1$ satisfying $\delta \le \beta <\frac{1+\delta }{2}$ 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 $n^{\frac{3}{4}}\le h(n)\le f(n)\le \sqrt{nh(n)/\log (n+1)}\le n^{\gamma }$ for some $\gamma <1$.