A multiplicatively symmetrized version of the Chung-Diaconis-Graham random process

TL;DR

This paper analyzes a symmetrized random process mod p, showing that about (log p)^2 steps are both necessary and sufficient for the process to become close to uniformly distributed.

Contribution

It introduces a multiplicatively symmetrized version of a Chung-Diaconis-Graham process and establishes precise mixing time bounds.

Findings

Order (log p)^2 steps suffice for mixing.

Order (log p)^2 steps are necessary for mixing.

The process achieves near-uniform distribution in quadratic logarithmic time.

Abstract

This paper considers random processes of the form $X_{n+1}=a_nX_n+b_n \pmod p$ where $p$ is odd, $X_0=0$, $(a_0,b_0), (a_1,b_1), (a_2,b_2),...$ are i.i.d., and $a_n$ and $b_n$ are independent with $P(a_n=2)=P(a_n=(p+1)/2)=1/2$ and $P(b_n=1)=P(b_n=0)=P(b_n=-1)=1/3$. This can be viewed as a multiplicatively symmetrized version of a random process of Chung, Diaconis, and Graham. This paper shows that order $(\log p)^2$ steps suffice for $X_n$ to be close to uniformly distributed on the integers mod $p$ for all odd $p$ while order $(\log p)^2$ steps are necessary for $X_n$ to be close to uniformly distributed on the integers mod $p$.