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July 2018 On the mixing time of Kac’s walk and other high-dimensional Gibbs samplers with constraints
Natesh S. Pillai, Aaron Smith
Ann. Probab. 46(4): 2345-2399 (July 2018). DOI: 10.1214/17-AOP1230

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 $O(n^{5}\log(n)^{2})$ in the preprint of Jiang [Jiang (2012)]. 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.

Citation

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Natesh S. Pillai. Aaron Smith. "On the mixing time of Kac’s walk and other high-dimensional Gibbs samplers with constraints." Ann. Probab. 46 (4) 2345 - 2399, July 2018. https://doi.org/10.1214/17-AOP1230

Information

Received: 1 June 2016; Revised: 1 May 2017; Published: July 2018
First available in Project Euclid: 13 June 2018

zbMATH: 06919027
MathSciNet: MR3813994
Digital Object Identifier: 10.1214/17-AOP1230

Subjects:
Primary: 60J10
Secondary: 60J20

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 4 • July 2018
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