Convergence of random walks on the circle generated by an irrational rotation

Abstract

Fix α ∈ [ 0 , 1 ) \alpha \in [0,1) . Consider the random walk on the circle S 1 S^1 which proceeds by repeatedly rotating points forward or backward, with probability 1 2 \frac 12 , by an angle 2 π α 2\pi \alpha . This paper analyzes the rate of convergence of this walk to the uniform distribution under “discrepancy” distance. The rate depends on the continued fraction properties of the number ξ = 2 α \xi =2\alpha . We obtain bounds for rates when ξ \xi is any irrational, and a sharp rate when ξ \xi is a quadratic irrational. In that case the discrepancy falls as k − 1 2 k^{-\frac 12} (up to constant factors), where k k is the number of steps in the walk. This is the first example of a sharp rate for a discrete walk on a continuous state space. It is obtained by establishing an interesting recurrence relation for the distribution of multiples of ξ \xi which allows for tighter bounds on terms which appear in the Erdős-Turán inequality.