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2022 Markov chains on finite fields with deterministic jumps
Jimmy He
Author Affiliations +
Electron. J. Probab. 27: 1-17 (2022). DOI: 10.1214/22-EJP757

Abstract

We study the Markov chain on Fp obtained by applying a function f and adding ±γ with equal probability. When f is a linear function, this is the well-studied Chung–Diaconis–Graham process. We consider two cases: when f is the extension of a non-linear rational function which is bijective, and when f(x)=x2. In the latter case, the stationary distribution is not uniform and we characterize it when p=3(mod4). In both cases, we give an almost linear bound on the mixing time, showing that the deterministic function dramatically speeds up mixing. The proofs involve establishing bounds on exponential sums over the union of short intervals.

Funding Statement

This research was supported in part by NSERC.

Acknowledgments

The author would like to thank Steve Butler, Alex Cowan, Persi Diaconis, and Kannan Soundararajan for their help. The author would also like to thank an anonymous referee for many suggestions which greatly improved the exposition.

Citation

Download Citation

Jimmy He. "Markov chains on finite fields with deterministic jumps." Electron. J. Probab. 27 1 - 17, 2022. https://doi.org/10.1214/22-EJP757

Information

Received: 24 June 2021; Accepted: 14 February 2022; Published: 2022
First available in Project Euclid: 25 February 2022

Digital Object Identifier: 10.1214/22-EJP757

Subjects:
Primary: 60J10
Secondary: 05C81 , 11T23

Keywords: Cheeger constant , Markov chain , mixing time , spectral gap

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