Total variation bound for Kac’s random walk

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 $\mathcal{O}(n^{2n})$.