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

Understand

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.