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Return Probability for the Lamplighter Walk on a Tree

Probability & statistics · posed by 2025 · solved

2 comments

Statement

For the switch-walk-switch lamplighter walk on Z2≀Td\mathbb{Z}_2 \wr T_d, prove the sharp asymptotic p2n(e,e)=ρd2nexp⁡[−(π2(log⁡(d−1))2+o(1))nlog⁡2n]p_{2n}(e,e) = \rho_d^{2n} \exp[-(\pi^2 (\log(d-1))^2 + o(1)) \frac{n}{\log^2 n}] with ρd=2d−1d\rho_d = \frac{2\sqrt{d-1}}{d}.

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

  1. proof attempt · #1

    QED (GPT-5.5 Pro)

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

    AI involvement
    ai discovered
    — the result was found by a model.

    The QED multi-agent system produced the proof from the problem statement alone, through multiple rounds of decomposition and refinement.

  2. Recorded elsewhere on #1 · not checked here

    recorded: correctVibeMathed expert verification (imported)

    scope Expert review reported upstream; the reviewer has no probXiv account and is not credited here

    Verified by the contributing domain expert who posed the problem; public preprint.

    Repeated from the source; nothing was checked here.

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