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Critical slowing down in the Sherrington–Kirkpatrick model

OpenAI · September 24, 2026

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Abstract

At the critical inverse temperature β = 1, we prove slow mixing for zero-field Gaussian Sherrington–Kirkpatrick heat-bath dynamics from typical equilibrium configurations held fixed as initial states. For every deterministic sequence $t_n=o(n^{2/3})$ in rate-one-per-site time, the Gibbs mass of initial states whose time-tn total-variation distance from equilibrium exceeds 1/4 tends to one in probability over the disorder. The same statement holds for every deterministic integer sequence $k_n=o(n^{5/3})$ of uniform-site update attempts.

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Comments and discussion

Formalization

  • OpenAI, Family 227 Lean scope (lean/docs/227.md, 2026-10-06). States that the release's Lean development covers this paper's three theorems: typical-start critical lower bounds, covariance and linear Rayleigh-quotient bounds of order n^{2/3}, and n^{2/3−ε} ≤ t_mix ≤ e^{εn} for worst-case starts (comparators CriticalSK.lean, CriticalSKMixing.lean). [family-level]

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Cite

@misc{OAI:Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026,
  author = {{OpenAI}},
  title = {{Critical slowing down in the Sherrington--Kirkpatrick model}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026/paper.pdf}{OAI:Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/227.md. The Lean statements have not been checked by people against the paper's claims.

Family 227: Critical SK autocorrelation processes and dynamics across the temperature transition

For zero-field Gaussian SK heat-bath dynamics with rate-one updates per spin, proves worst-start cutoff on the $\log n$ scale for fixed $0\le\beta\lt 1$, mixing time $n^{2/3+o(1)}$ at β = 1, and stretched-exponential mixing from a Gibbs-sampled fixed starting configuration for β > 1, in probability over disorder. At criticality, rescaled stationary and quench autocorrelation processes have universal random limits for Gaussian and Rademacher disorder; the quench limit relaxes to the stationary limit.

Other papers in this family: