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Talagrand's critical Sherrington-Kirkpatrick overlap conjecture

Mathematical physics · posed by Michel Talagrand, 2011 · solved

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Statement

At the critical inverse temperature β=1\beta=1 in the Sherrington-Kirkpatrick spin glass model, Talagrand conjectured that the expected squared overlap of two independent Gibbs replicas has an exact N−2/3N^{-2/3} scaling: there exists a constant a>0a>0 such that

lim⁡N→∞N2/3E⟨R1,22⟩=a.\lim_{N\to\infty} N^{2/3}\mathbb{E}\langle R_{1,2}^2\rangle=a.

Du and Huang prove that this limit exists and is positive and finite. More strongly, they determine the full limiting quenched distribution of the rescaled overlap N1/3R1,2N^{1/3}R_{1,2} in terms of the reflected Airy1\mathrm{Airy}_1 point process.

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

  1. proof attempt · #1

    Hang Du and Brice Huang, using GPT-5.6 Pro

    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 authors state that most of the arguments in the paper were generated using GPT-5.6 Pro, with the goal of exploring further consequences of ideas developed in their companion work on critical SK free-energy fluctuations. The paper does not assign individual lemmas or the central comparison principle specifically to the model, so the contribution is classified conservatively as AI co-developed rather than AI discovered.

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