Kac's walk on n-sphere mixes in nlogn steps

Determining the mixing time of Kac's random walk on the sphere Sn-1 is a long-standing open problem. We show that the total variation mixing time of Kac's walk on Sn-1 is between $\frac{1}{2}n\log \left( n \right)$ and 200n log(n) for all n sufficiently large. Our bound is thus optimal up to a constant factor, improving on the best-known upper bound of O(n. log(n)²) due to Jiang [Ann. Appl. Probab. 22 (2012) 1712-1727]. Our main tool is a "non-Markovian" coupling recently introduced by the second author in [Ann. Appl. Probab. 24 (2014) 114-130] for obtaining the convergence rates of certain high dimensional Gibbs samplers in continuous state spaces.