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Publication 23-CNA-003

Using Bernoulli Maps to Accelerate Mixing of a Random Walk on the Torus

Gautam Iyer
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213

Ethan Lu
Department of Mathematics
Stanford University
Stanford, CA 94305

James Nolen
Department of Mathematics
Duke University
243 Physics Building, Durham, NC 27708

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 is $O$(1/$\epsilon^2$), where $\epsilon$ is the step size. Our main results show that for a class of Bernoulli maps, when the random walk is alternated with the Bernoulli map $\phi$ the mixing time becomes $O$(|ln $\epsilon$|). We also study the dissipation time of this process, and obtain $O$(|ln $\epsilon$|) upper and lower bounds with explicit constants.

Dedicated to Robert L. Pego, whose life and work is an inspiration.

Get the paper in its entirety as  23-CNA-003.pdf

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