Total variation cutoff for Kac's walk on the sphere
arXiv:2607.13401
summary
The paper proves that the discrete-time Kac walk on the (n‑1)-dimensional sphere, started from a coordinate vector, exhibits a total‑variation cutoff at time C_{BRW}·n·log n (with C_{BRW}≈3.8916), showing the cutoff occurs earlier than the previously conjectured 2n·log n.
Abstract
We prove cutoff in total variation distance for the discrete-time Kac walk on started from a coordinate vector. The cutoff occurs at , where 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 .
Topics & keywords
#mixing time#cutoff phenomenon#kac walk#total variation distance#branching random walk#high-dimensional spheretotal variation distanceKac walkcutoffbranching random walkmixing timeS^{n-1}