Mixing time of the Chung–Diaconis–Graham random process

Abstract

Define $(X_n)$ on ${\mathbf {Z}}/q{\mathbf {Z}}$ by $X_{n+1} = 2X_n + b_n$, where the steps $b_n$ are chosen independently at random from $-1, 0, +1$. The mixing time of this random walk is known to be at most $1.02 \log _2 q$ for almost all odd q (Chung, Diaconis, Graham in Ann Probab 15(3):1148–1165, 1987), and at least $1.004 \log _2 q$ (Hildebrand in Proc Am Math Soc 137(4):1479–1487, 2009). We identify a constant $c = 1.01136\dots$ such that the mixing time is $(c+o(1))\log _2 q$ for almost all odd q. In general, the mixing time of the Markov chain $X_{n+1} = a X_n + b_n$ modulo q, where a is a fixed positive integer and the steps $b_n$ are i.i.d. with some given distribution in ${\mathbf {Z}}$, is related to the entropy of a corresponding self-similar Cantor-like measure (such as a Bernoulli convolution). We estimate the mixing time up to a $1+o(1)$ factor whenever the entropy exceeds $(\log a)/2$.