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
gautam@math.cmu.edu

Ethan Lu
Department of Mathematics
Stanford University
Stanford, CA 94305
ethanlu@stanford.edu

James Nolen
Department of Mathematics
Duke University
243 Physics Building, Durham, NC 27708
nolen@math.duke.edu

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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