Accelerating abelian random walks with hyperbolic dynamics

Abstract

Given integers $$d \ge 2, n \ge 1$$ , we consider affine random walks on torii $$(\mathbb {Z}/ n \mathbb {Z})^{d}$$ defined as $$X_{t+1} = A X_{t} + B_{t} \mod n$$ , where $$A \in \mathrm {GL}_{d}(\mathbb {Z})$$ is a invertible matrix with integer entries and $$(B_{t})_{t \ge 0}$$ is a sequence of iid random increments on $$\mathbb {Z}^{d}$$ . We show that when A has no eigenvalues of modulus 1, this random walk mixes in $$O(\log n \log \log n)$$ steps as $$n \rightarrow \infty $$ , 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 $$x \mapsto A^{\top } x$$ on the continuous torus $$\mathbb {R}^{d} / \mathbb {Z}^{d}$$ . 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.