Total-variation mixing of the k-column Kac walk

Extend the mixing result for the k-column walk of Kac’s walk on the Stiefel manifold from Wasserstein distance to total variation distance.

Background

The paper proves that the projection of Kac’s walk onto its first k columns mixes rapidly in Wasserstein distance, with a bound uniform in n and k. The authors explicitly ask whether this stronger probabilistic convergence norm can be obtained in total variation distance as well, identifying this as a plausible direction for future work.