Accelerating Abelian Random Walks with Hyperbolic Dynamics

TL;DR

This paper demonstrates that affine random walks on high-dimensional tori with hyperbolic dynamics mix rapidly, in logarithmic time, leveraging properties of chaotic automorphisms to accelerate convergence.

Contribution

It extends mixing time results to higher dimensions using hyperbolic automorphisms, generalizing previous one-dimensional findings and connecting chaotic dynamics with Markov chain mixing.

Findings

Mixing in O(log n) steps for almost all n.

Mixing in O(log n log log n) steps generally.

Hyperbolic automorphisms accelerate mixing speed.

Abstract

Given integers $d \geq 2, n \geq 1$, 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 an invertible matrix with integer entries and $(B_{t})_{t \geq 0}$ 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 generalize those 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.