Total variation cutoff for Kac's walk on the sphere

Vishesh Jain; Clayton Mizgerd

Total variation cutoff for Kac's walk on the sphere

Probability 2026-07-15 v1 Discrete Mathematics

Authors: Vishesh Jain , Clayton Mizgerd

Abstract

We prove cutoff in total variation distance for the discrete-time Kac walk on $S^{n-1}$ started from a coordinate vector. The cutoff occurs at $C_{\mathrm{BRW}}n\log n$ , where $C_{\mathrm{BRW}} \approx 3.8916$ is an explicit constant determined by the speed of the leftmost particle in a branching random walk. In particular, the cutoff location is not at the conjectured time $2n\log n$ .

Keywords

total variation cutoff Kac walk sphere branching random walk mixing time

Cite

@article{arxiv.2607.13401,
  title  = {Total variation cutoff for Kac's walk on the sphere},
  author = {Vishesh Jain and Clayton Mizgerd},
  journal= {arXiv preprint arXiv:2607.13401},
  year   = {2026}
}