Markov chains on finite fields with deterministic jumps

Abstract

We study the Markov chain on $\mathbf{F}_p$ obtained by applying a function $f$ and adding $\pm\gamma$ 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 rational function which is bijective, and when $f(x)=x^2$. In the latter case, the stationary distribution is not uniform and we characterize it when $p=3\pmod{4}$. 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.