The speed of random walks on semigroups

Abstract

We construct, for each real number $0\le \hat{I}\pm \le 1$, a random walk on a finitely generated semigroup whose speed exponent is $\hat{I}\pm$. We further show that the speed function of a random walk on a finitely generated semigroup can be arbitrarily slow, yet tending to infinity. These phenomena demonstrate a sharp contrast from the group-theoretic setting. On the other hand, we show that the distance of a random walk on a finitely generated semigroup from its starting position is infinitely often larger than a non-constant universal lower bound, excluding a certain degenerate case.

22 pages, 1 figure