Random walks on the circle and Diophantine approximation

Berkes, István and Borda, Bence (2023) Random walks on the circle and Diophantine approximation. JOURNAL OF THE LONDON MATHEMATICAL SOCIETY, 108 (2). pp. 409-440. ISSN 0024-6107

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Abstract

Random walks on the circle group R/Z whose elementary steps are lattice variables with span alpha is not an element of Q or p/q is an element of Q taken mod Z exhibit delicate behavior. In the rational case, we have a random walk on the finite cyclic subgroup 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 -> 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 phase transition from polynomial to exponential decay after approximate to 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.

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