On The Convergence to Equilibrium of Kac's Random Walk on Matrices

Roberto Imbuzeiro Oliveira

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 x and y (resp.) that contracts distances by a (ξ + o(1)) factor on average. Then the map μ ↦ μ P is ξ-contracting in the transportation cost metric.