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June 2009 On the convergence to equilibrium of Kac’s random walk on matrices
Roberto Imbuzeiro Oliveira
Ann. Appl. Probab. 19(3): 1200-1231 (June 2009). DOI: 10.1214/08-AAP550

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(n2ln n) steps. We also prove that our bound is at most a O(ln 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, yM 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 μμP is ξ-contracting in the transportation cost metric.

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Roberto Imbuzeiro Oliveira. "On the convergence to equilibrium of Kac’s random walk on matrices." Ann. Appl. Probab. 19 (3) 1200 - 1231, June 2009. https://doi.org/10.1214/08-AAP550

Information

Published: June 2009
First available in Project Euclid: 15 June 2009

zbMATH: 1173.60343
MathSciNet: MR2537204
Digital Object Identifier: 10.1214/08-AAP550

Subjects:
Primary: 60J27
Secondary: 65C40

Keywords: Kac’s random walk , Markov chain , mixing time , path coupling

Rights: Copyright © 2009 Institute of Mathematical Statistics

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Vol.19 • No. 3 • June 2009
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