Cutoff for Kac's Walk on the Sphere
Prior state unknown→proved
the conjectured cutoff location of 2n log n is wrong
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probability-statistics / Markov chains
Cutoff for Kac's Walk on the Sphere
The discrete-time Kac walk on $S^{n-1}$ started from a coordinate vector exhibits total variation cutoff at $C_{\mathrm{BRW}} n \log n$, where $C_{\mathrm{BRW}} \approx 3.8916$ is set by the speed of the leftmost particle in a branching random walk. The cutoff is therefore not at the conjectured $2n\log n$.
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The discrete-time Kac walk on $S^{n-1}$ started from a coordinate vector exhibits total variation cutoff at $C_{\mathrm{BRW}} n \log n$, where $C_{\mathrm{BRW}} \approx 3.8916$ is set by the speed of the leftmost particle in a branching random walk. The cutoff is therefore not at the conjectured $2n\log n$.
the conjectured cutoff location of 2n log n is wrong
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