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

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.