Cut-off phenomenon for the ax+b Markov chain over a finite field

TL;DR

This paper investigates the mixing times of a linear Markov chain over finite fields, demonstrating a sharp cut-off phenomenon under certain number-theoretic assumptions and providing unconditional bounds.

Contribution

It establishes the occurrence of the cut-off phenomenon for the ax+b Markov chain over finite fields under the Riemann hypothesis, with unconditional bounds also derived.

Findings

Cut-off phenomenon occurs for most primes and values of a under RH.

Unconditional upper bounds for mixing time are provided.

Results connect number theory with Markov chain mixing behavior.

Abstract

We study the Markov chain $x_{n+1}=ax_n+b_n$ on a finite field $\mathbb{F}_p$, where $a \in \mathbb{F}_p$ is fixed and $b_n$ are independent and identically distributed random variables in $\mathbb{F}_p$. Conditionally on the Riemann hypothesis for all Dedekind zeta functions, we show that the chain exhibits a cut-off phenomenon for most primes $p$ and most values of $a \in \mathbb{F}_p$. We also obtain weaker, but unconditional, upper bounds for the mixing time.