On the convergence to equilibrium of Kac's random walk on matrices

Abstract

We consider Kac's random walk on n-dimensional rotation matrices, where each step is a random rotation in the plane generated by two randomly picked coordinates. We show that this process converges to the Haar measure on SO(n) in the L2 transportation cost (Wasserstein) metric in O(n2 lnn) steps. We also prove that our bound is at most a O(In n) factor away from optimal. Previous bounds, due to Diaconis/Saloff-Coste and Pak/Sidenko, had extra powers of n and held only for L1 transportation cost. Our proof method includes a general result of independent interest, akin to the path coupling method of Bubley and Dyer. Suppose that P is a Markov chain on a Polish length space (M, d) and that for all x, y ∈ M with d(x, y) «1 there is a coupling (X, Y) of one step of P from ∼ and y (resp.) that contracts distances by a (ζ + o(1)) factor on average. Then the map μ → μP is ζ-contracting in the transportation cost metric. © Institute of Mathematical Statistics, 2009.