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Mixing Time of Kac's Walk on the Rotation Group

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kac-walk-rotation-mixingProbability & statisticsposed by Walk introduced by W. K. Hastings; optimal rate the standing target of the mixing-time literature, 1970recorded: solved

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Statement

Kac's walk on the rotation group, introduced by Hastings in 1970, is a central high-dimensional Markov chain in statistical physics and computational science. The paper proves it mixes in n2log⁡nn^2 \log n steps, the conjectured optimal rate, closing the gap left by a long line of successive improvements.

Context

The mixing time of Kac's walk on SO(n) has been a standing target of the Markov chain community since Hastings (1970), with a documented ladder of bounds by Diaconis-Saloff-Coste, Pillai-Smith and others.

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  • #1

    Attempt 1

    proof attemptGPT Pro 5.4 with Natesh S. Pillai, Aaron Smith ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    GPT Pro 5.4
    people
    Natesh S. Pillai, Aaron Smith

    "Much of this paper was developed in collaboration with GPT Pro 5.4. In particular, the idea of using matrix martingale inequalities" came from that collaboration. Pillai and Smith are the authors of the previous best bounds for this walk.

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