Markov chains on finite fields with deterministic jumps

TL;DR

This paper investigates Markov chains on finite fields driven by a function and random jumps, analyzing mixing times and stationary distributions for specific functions, including linear and quadratic cases.

Contribution

It characterizes the stationary distribution for quadratic functions and provides bounds on mixing times, highlighting the impact of deterministic functions on mixing speed.

Findings

Mixing time is almost linear in the size of the field.

Stationary distribution is non-uniform for quadratic functions when p ≡ 3 mod 4.

Deterministic functions significantly accelerate mixing.

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.