Accelerating abelian random walks with hyperbolic dynamics

Bastien Dubail, Laurent Massoulié

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 ∈ GLd(Z) is a invertible matrix with integer entries and ( Bt )t ≥0 is a sequence of iid random increments on Zd . 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 are similar to those of Chung et al. (Ann Probab 15(3):1148–1165, 1987) 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 Rd /Zd . 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.