# Random walks on the circle and Diophantine approximation

## TL;DR

This paper studies random walks on a circle and how they behave differently when the step size is rational or irrational.

## Contribution

The paper introduces a new phenomenon in random walks on compact groups involving phase transitions in convergence rates.

## Key findings

- Random walks with irrational span exhibit different behavior compared to rational spans.
- A phase transition from polynomial to exponential decay occurs in the rational case after ≈q² steps.
- Convergence rates differ between the Kolmogorov and total variation metrics.

## Abstract

Random walks on the circle group R/Z whose elementary steps are lattice variables with span α∉Q or p/q∈Q taken mod Z exhibit delicate behavior. In the rational case, we have a random walk on the finite cyclic subgroup Zq, 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 α, and explicitly describe the transition of these Markov chains from finite to general state space as p/q→α 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 phase transition from polynomial to exponential decay after ≈q2 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.