We study the Markov chain on obtained by applying a function f and adding 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 . In the latter case, the stationary distribution is not uniform and we characterize it when . 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.
This research was supported in part by NSERC.
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.
"Markov chains on finite fields with deterministic jumps." Electron. J. Probab. 27 1 - 17, 2022. https://doi.org/10.1214/22-EJP757