Random walks on the circle and Diophantine approximation

TL;DR

This paper studies the behavior of random walks on the circle group, extending classical results from rational to irrational steps, and reveals new phenomena in convergence rates related to Diophantine approximation.

Contribution

It extends classical finite state Markov chain results to irrational steps on the circle and describes the transition from finite to infinite state behavior as rational approximations improve.

Findings

Classical CLT and LIL hold for rational steps on finite cyclic groups.

Transition from finite to general state space as rational approximations approach irrational spans.

Surprising polynomial to exponential decay transition in convergence rates after about q^2 steps.

Abstract

Random walks on the circle group $\mathbb{R}/\mathbb{Z}$ whose elementary steps are lattice variables with span $\alpha \not\in \mathbb{Q}$ or $p/q \in \mathbb{Q}$ taken mod $\mathbb{Z}$ exhibit delicate behavior. In the rational case we have a random walk on the finite cyclic subgroup $\mathbb{Z}_q$, and the central limit theorem and the law of the iterated logarithm follow from classical results on finite state space Markov chains. In this paper we extend these results to random walks with irrational span $\alpha$, and explicitly describe the transition of these Markov chains from finite to general state space as $p/q \to \alpha$ along the sequence of best rational approximations. We also consider the rate of weak convergence to the stationary distribution in the Kolmogorov metric, and in the rational case observe a surprising transition from polynomial to exponential decay after $\approx q^2$ steps; this seems to be a new phenomenon in the theory of random walks on compact groups. In contrast, the rate of weak convergence to the stationary distribution in the total variation metric is purely exponential.