Total variation bound for Kac's random walk

Abstract

We show that the classical Kac's random walk on (n-1)-sphere S n-1 starting from the point mass at e1 mixes in O(n 5(log n)3) steps in total variation distance. The main argument uses a truncation of the running density after a burn-in period, followed by L2 convergence using the spectral gap information derived by other authors. This improves upon a previous bound by Diaconis and Saloff-Coste of order O(n2n). © 2012 Institute of Mathematical Statistics.