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Critical mixing in the Sherrington–Kirkpatrick model

OpenAI · September 25, 2026

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Abstract

At the critical inverse temperature β = 1, we determine the worst-start total-variation mixing exponent for single-site heat-bath dynamics in the zero-field Gaussian Sherrington–Kirkpatrick model. The mixing time is $n^{2/3+o(1)}$ when every site has rate one, or $n^{5/3+o(1)}$ attempted uniform-site updates, in probability over the disorder.

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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Comments and discussion

Formalization

  • OpenAI, Family 227 Lean scope (OpenAI math release, 2026-10-06). The family formalization includes the earlier critical polynomial lower and subexponential upper mixing bounds, rather than this companion’s matching $2/3$ upper exponent. [family-level]

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Cite

@misc{OAI:Critical-mixing-in-the-Sherrington-Kirkpatrick-model-September-25-2026,
  author = {{OpenAI}},
  title = {{Critical mixing in the Sherrington--Kirkpatrick model}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Critical-mixing-in-the-Sherrington-Kirkpatrick-model-September-25-2026/paper.pdf}{OAI:Critical-mixing-in-the-Sherrington-Kirkpatrick-model-September-25-2026}},
  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: