Total variation cutoff for Kac's walk on the sphere

2026-07-14

AI Summary

This study investigates the mixing behavior in total variation distance of a discrete-time random walk on the Kac sphere, initiated from a coordinate vector. By integrating probabilistic coupling techniques, spectral analysis, and limit theory for branching random walks, the authors rigorously establish—for the first time—the existence of a cutoff phenomenon for this model. They precisely identify the cutoff time as \( C_{\text{BRW}} \cdot n \log n \), where \( C_{\text{BRW}} \approx 3.8916 \) is determined by the speed of the leftmost particle in an associated branching random walk. This finding refutes a prior conjecture that the cutoff occurs at \( 2n \log n \), thereby substantially revising and refining the understanding of the mixing time for this stochastic process.

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$.

Problem

• total variation cutoff
• Kac's walk
• mixing time
• sphere
• branching random walk

Innovation

• total variation cutoff
• Kac's walk
• branching random walk
• mixing time
• sphere