Mixing time of the Chung--Diaconis--Graham random process
Abstract
Define (Xn) on Z/qZ by Xn+1 = 2Xn + bn, where the steps bn are chosen independently at random from -1, 0, +1. The mixing time of this random walk is known to be at most 1.02 2 q for almost all odd q (Chung--Diaconis--Graham, 1987), and at least 1.004 2 q (Hildebrand, 2008). We identify a constant c = 1.01136… such that the mixing time is (c+o(1))2 q for almost all odd q. In general, the mixing time of the Markov chain Xn+1 = a Xn + bn modulo q, where a is a fixed positive integer and the steps bn are i.i.d. with some given distribution in 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 ( a)/2.
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