Kac's Walk on $n$-sphere mixes in $n\log n$ steps

TL;DR

This paper establishes that Kac's walk on the n-sphere mixes in a time proportional to n log n, providing nearly tight bounds and improving previous results significantly.

Contribution

It proves the mixing time of Kac's walk is between n log n/2 and 200 n log n, nearly matching the conjectured optimal rate and improving prior bounds.

Findings

Mixing time is between 0.5 n log n and 200 n log n.

Improves previous upper bound from O(n^5 log n^2).

Uses a novel non-Markovian coupling technique.

Abstract

Determining the mixing time of Kac's random walk on the sphere $\mathrm{S}^{n-1}$ is a long-standing open problem. We show that the total variation mixing time of Kac's walk on $\mathrm{S}^{n-1}$ is between $\frac{1}{2} \, n \log(n)$ and $200 \,n \log(n)$. Our bound is thus optimal up to a constant factor, improving on the best-known upper bound of $O(n^{5} \log(n)^{2})$ due to Jiang. Our main tool is a `non-Markovian' coupling recently introduced by the second author for obtaining the convergence rates of certain high dimensional Gibbs samplers in continuous state spaces.