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Fast mixing of Glauber dynamics for the Sherrington-Kirkpatrick model throughout the high-temperature phase, zero external field

For the Sherrington-Kirkpatrick model with couplings Jij∼N(0,β2/n)J_{ij}\sim N(0,\beta^2/n), single-site Glauber (heat-bath) dynamics resamples one spin from its conditional Gibbs law. Physics predicts fast relaxation in the whole replica-symmetric phase β<1\beta<1; rigorous fast-mixing results reached β<1/4\beta<1/4 (Eldan-Koehler-Zeitouni; Anari et al.), about 0.2950.295 (Anari-Koehler-Vuong), β<1/2\beta<1/2 (Wang 2026) and 1/2+ε01/2+\varepsilon_0 (Boban-Li-Oveis Gharan 2026). Open Problem 15 of the ETH randomstrasse101 list asks: considering the SK model with an arbitrary external field, does Glauber dynamics fast mix (in polynomial time) up to the replica symmetry breaking threshold β∗=1\beta^*=1?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Spin glasses, Markov chain mixing
Posed by
Almut Roedder (ETH randomstrasse101 open problems, Problem 15)
Year posed
2024
Years open
2y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
30 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Spectral-gap paper, Theorem 1.1: for fixed 0<β<10<\beta<1, with probability tending to one over the disorder, Varμ0(f)≤CβD0(f)\mathrm{Var}_{\mu_0}(f)\le C_\beta\mathcal D_0(f) for all ff (zero field), so the uniform-site heat-bath chain has gap at least 1/(Cβn)1/(C_\beta n). Companions: worst-start total-variation cutoff at log⁡n/(2λ(β))\log n/(2\lambda(\beta)) for 0≤β<10\le\beta<1; at β=1\beta=1 mixing time n2/3+o(1)n^{2/3+o(1)} and slow mixing from typical starts before n2/3n^{2/3}; for β>1\beta>1 stretched-exponential barriers from typical starts and an exp⁡(n1−c)\exp(n^{1-c}) typical-start upper bound; at β=1\beta=1 universal random limits of rescaled autocorrelations for Gaussian and Rademacher disorder. Not shown: nonzero external fields, or an O(nlog⁡n)O(n\log n) bound from the gap paper alone.

What the AI did

The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family has eight manuscripts (September 24 to October 5, 2026): the spectral gap paper (principal here), worst-start cutoff for beta < 1, critical slowing down and the critical mixing exponent n^(2/3) at beta = 1, typical-start lower and upper bounds for beta > 1, and two papers on universal critical autocorrelation limits.

Verification

No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 of the spectral-gap paper were read against Open Problem 15. Scope limit: zero external field only, while the problem asks for arbitrary fields; polynomial mixing follows from the gap by the standard bound (about n2n^2 updates), not stated as a separate theorem there. Lean: the challenge SKHighTemperature (theorem OAI.SKGap.sk_main, solution module OAI.Probability.SKGap.Main) exists at the pinned commit but is not in the formalization catalogue; its statement was read here: for each 0<β<10<\beta<1 a constant CC such that the zero-field Poincare inequality with the heat-bath Dirichlet form, and discrete gap at least 1/(Cn)1/(Cn), hold with disorder probability tending to one. That is the principal theorem. Not rebuilt here. Other Lean challenges (CriticalSK, CriticalSKMixing, SKRatio, SKBarriers) cover companion results; SKRatio covers only beta < 1/2.

Sources

Changelog1 change

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