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Cutoff for Kac's Walk on the Sphere

Probability & statistics · posed by Mark Kac, 1956 · solved

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Statement

The discrete-time Kac walk on Sn−1S^{n-1} started from a coordinate vector exhibits total variation cutoff at CBRWnlog⁡nC_{\mathrm{BRW}} n \log n, where CBRW≈3.8916C_{\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 2nlog⁡n2n\log n.

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. proof attempt · #1

    Vishesh Jain and Clayton Mizgerd, using ChatGPT

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

    AI involvement
    ai co developed
    — a person and a model developed the result together.

    The acknowledgement says the authors used ChatGPT extensively, at the level of a co-author, for brainstorming, help with proofs, literature review, checking for mistakes, writing code, and preparing the manuscript. It does not separate which parts came from where.

    the conjectured cutoff location of 2n log n is wrong

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Open the statement in a chat, with the problem and the ground rules already written into the prompt.

open in Claude →open in ChatGPT →

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