JournalsggdVol. 11, No. 2pp. 455–467

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

  • Gideon Amir

    Bar-Ilan University, Ramat Gan, Israel
On the joint behaviour of speed and entropy of random walks on groups cover
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Abstract

For every 3/4δ,β<13/4 \le \delta, \beta< 1 satisfying δβ<1+δ2\delta\leq \beta < \frac{1+\delta}{2} we construct a finitely generated group Γ\Gamma and a (symmetric, finitely supported) random walk XnX_n on Γ\Gamma so that its expected distance from its starting point satisfies EXnnβ\mathbf E |X_n|\asymp n^{\beta} and its entropy satisfies H(Xn)nδH(X_n)\asymp n^\delta. In fact, the speed and entropy can be set precisely to equal any two nice enough prescribed functions f,hf,h up to a constant factor as long as the functions satisfy the relation n34h(n)f(n)nh(n)/log(n+1)nγn^{\frac{3}{4}}\leq h(n)\leq f(n)\leq \sqrt{{nh(n)}/{\log (n+1)}}\leq n^\gamma for some γ<1\gamma<1.

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. 455–467

DOI 10.4171/GGD/403