Accelerating Abelian Random Walks with Hyperbolic Dynamics

Abstract

Given integers d ≥ 2, n ≥ 1, we consider affine random walks on torii (Z nZ) d defined as Xt+1 = AXt + Bt mod n, where A ∈ GL d (Z) is an invertible matrix with integer entries and (Bt)t≥0 is a sequence of iid random increments on Z d . We show that when A has no eigenvalues of modulus 1, this random walk mixes in O(log n log log n) steps as n → ∞, and mixes actually in O(log n) steps only for almost all n. These results generalize those of [11] on the so-called Chung-Diaconis-Graham process, which corresponds to the case d = 1. Our proof is based on the initial arguments of Chung, Diaconis and Graham, and relies extensively on the properties of the dynamical system x ↦ A ⊺ x on the continuous torus R d Z d . Having no eigenvalue of modulus one makes this dynamical system a hyperbolic toral automorphism, a typical example of a chaotic system known to have a rich behaviour. As such our proof sheds new light on the speed-up gained by applying a deterministic map to a Markov chain.