ON THE CONVERGENCE TO EQUILIBRIUM OF KAC'S RANDOM WALK ON MATRICES

• 成果类型: Article
• 署名作者: Oliveira, Roberto Imbuzeiro
• 署名单位: Instituto Nacional de Matematica Pura e Aplicada (IMPA)
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/08-AAP550
• 发表日期: 2009
• 页码: 1200-1231
• 关键词: metric-measure-spaces; master equation; RICCI CURVATURE; spectral gap; transport; geometry

摘要

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 L-2 transportation cost (Wasserstein) metric in O(n(2) In n) 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 L-1 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 is an element of 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 (xi + o(1)) factor on average. Then the map mu -> mu P is xi-contracting in the transportation cost metric.