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

Iyer, Gautam; Lu, Ethan; Nolen, James

doi:10.1090/qam/1668

Authors: Gautam Iyer, Ethan Lu and James Nolen
Journal: Quart. Appl. Math. 82 (2024), 359-390
MSC (2020): Primary 60J05; Secondary 37A25
DOI: https://doi.org/10.1090/qam/1668
Published electronically: June 8, 2023
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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 $O(1/\varepsilon ^2)$, where $\varepsilon$ 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 $\varphi$ the mixing time becomes $O(\lvert \ln \varepsilon \rvert )$. We also study the dissipation time of this process, and obtain $O(\lvert \ln \varepsilon \rvert )$ upper and lower bounds with explicit constants.

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