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

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

2007-05-15

arxiv.org/abs/0705.2253v4

Annals of Applied Probability 2009, Vol. 19, No. 3, 1200-1231

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/08-AAP550 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

10.1214/08-AAP550

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 $\mathit{SO}(n)$ in the $L^2$ transportation cost (Wasserstein) metric in $O(n^2\ln n)$ 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 $L^1$ 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 $x,y\in M$ with $d(x,y)\ll1$ 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 $\mu\mapsto\mu P$ is $\xi$-contracting in the transportation cost metric.

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