On the mixing time of Kac’s walk and other high-dimensional Gibbs samplers with constraints

Abstract

Determining the total variation mixing time of Kac's random walk on the special orthogonal group SO(n) has been a long-standing open problem. In this paper, we construct a novel non-Markovian coupling for bounding this mixing time. The analysis of our coupling entails controlling the smallest singular value of a certain random matrix with highly dependent entries. The dependence of the entries in our matrix makes it not-amenable to existing techniques in random matrix theory. To circumvent this difficulty, we extend some recent bounds on the smallest singular values of matrices with independent entries to our setting. These bounds imply that the mixing time of Kac's walk on the group SO(n) is between C 1 n 2 and C 2 n 4 log(n) for some explicit constants 0 < C 1 , C 2 < ∞, substantially improving on the bound of O(n 5 log(n) 2 ) in the preprint [Jia12b] of Jiang. Our methods may also be applied to other high dimensional Gibbs samplers with constraints and thus are of independent interest. In addition to giving analytical bounds on the mixing time, our approach allows us to compute rigorous estimates of the mixing time by simulating the eigenvalues of a random matrix. ‡

New tests of uniformity on the compact classical groups as diagnostics for weak-$^{*}$ mixing of Markov chains

“…Convergence of the Kac's random walk has been studied by various authors in different senses. In the current discussion, the focus is on convergence in Wasserstein distance which metrizes the weak convergence; for a review of the literature see Pak and Sidenko (2007); Oliveira (2009); Pillai and Smith (2016). The best known bound on the mixing-time in Wasserstein distance is obtained by Oliveira (2009), providing an upper bound of order n 2 log n on the mixing-time which is at most a factor log n away from optimal.…”