Using Bernoulli maps to accelerate mixing of a random walk on the torus

Abstract

We study the mixing time of a random walk on the torus, alternated with a Lebesgue measure preserving Bernoulli map. Without the Bernoulli map, the mixing time of the random walk alone isO(1/ε2)O(1/\varepsilon ^2), whereε\varepsilonis the step size. Our main results show that for a class of Bernoulli maps, when the random walk is alternated with the Bernoulli map φ\varphithe mixing time becomesO(|ln⁡ε|)O(\lvert \ln \varepsilon \rvert ). We also study thedissipation timeof this process, and obtain O(|ln⁡ε|)O(\lvert \ln \varepsilon \rvert )upper and lower bounds with explicit constants.