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.

References

Can the $k$-column mixing result be extended to TV distance and not just Wasserstein? This appears plausible, and we leave it for future work.

— On the Pseudo-Mixing of Kac's Walk  (2608.17374 - Pillai et al., 18 Aug 2026) in Section “Open Questions,” Subsection “Immediate Open Questions”

Although our main theorem is stated for the complex Stiefel manifold, our method should also apply to the standard Kac’s walk on the real Stiefel manifold, where each step applies a random rotation to a single pair of coordinates. This would yield a total variation mixing time of O(d(k+\log d)\log(d/\varepsilon)) sequential steps, matching the Wasserstein mixing bound established in . Such a result would resolve their open question of whether the k-column walk satisfies the same mixing bound in total variation distance.

— Rapid Mixing of Parallel Kac's Walk: From Spheres to Stiefel Manifolds  (2609.37999 - Chen et al., 29 Sep 2026) in Section 1, subsection “Discussion”