On the Mixing Time of Kac's Walk and Other High-Dimensional Gibbs Samplers with Constraints

TL;DR

This paper establishes new bounds on the mixing time of Kac's walk on the special orthogonal group, using novel couplings and random matrix theory extensions, improving previous bounds significantly.

Contribution

Introduces a non-Markovian coupling approach and extends random matrix bounds to analyze the mixing time of high-dimensional Gibbs samplers with constraints.

Findings

Mixing time bounds between C₁ n² and C₂ n⁴ log(n).

New coupling method for analyzing Kac's walk.

Method applicable to other constrained Gibbs samplers.

Abstract

Determining the total variation mixing time of Kac's random walk on the special orthogonal group $\mathrm{SO}(n)$ 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 $\mathrm{SO}(n)$ is between $C_{1} n^{2}$ and $C_{2} n^{4} \log(n)$ for some explicit constants $0 < C_{1}, C_{2} < \infty$, substantially improving on the bound of $O(n^{5} \log(n)^{2})$ by 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.