Entropy, isoperimetry and the total variation profile of random walks

Abstract

The total variation profile $\mathrm{TV}_n$ of a random walk measures how far its distribution at time $n$ is from being invariant under the generators. Three lower bounds are given: an elementary bound $\mathrm{TV}_n \geq 1/\big(44\,L(n)\big)$ in terms of the speed $L(n)$; a bound by the entropy increment $H(P^{n+1}) - H(P^n)$, which is a reverse Pinsker inequality; and a bound in terms of the entropy and the isoperimetric profile. Deciding whether the entropy-increment bound always improves on the isoperimetric one is related to conjectures on the growth of the entropy and of groups. The motivation comes from the question of whether the distributions of a random walk can be transported back onto their translates at bounded cost.