Total variation bound for Kac's random walk

TL;DR

This paper establishes a polynomial mixing time bound for Kac's random walk on the sphere, significantly improving previous exponential bounds by employing spectral gap analysis and density truncation techniques.

Contribution

It provides the first polynomial upper bound on the mixing time of Kac's walk, advancing understanding of its convergence rate.

Findings

Mixing time is O(n^5 (log n)^3) steps in total variation distance.

Improves previous bound from exponential to polynomial in n.

Uses spectral gap and density truncation methods for analysis.

Abstract

We show that the classical Kac's random walk on $(n-1)$-sphere $S^{n-1}$ starting from the point mass at $e_1$ mixes in $\mathcal{O}(n^5(\log n)^3)$ steps in total variation distance. The main argument uses a truncation of the running density after a burn-in period, followed by $\mathcal{L}^2$ convergence using the spectral gap information derived by other authors. This improves upon a previous bound by Diaconis and Saloff-Coste of order $\mathcal {O}(n^{2n})$.