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A typical-start upper bound for low-temperature SK Glauber dynamics

OpenAI · September 24, 2026

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Abstract

For every fixed inverse temperature β > 1, we prove subexponential mixing for rate-one-per-site heat-bath dynamics in the zero-field Gaussian Sherrington–Kirkpatrick model from a typical equilibrium configuration. If one Gibbs-sampled configuration is held fixed as the initial state, its total-variation distance from equilibrium is at most 1/4 by time $\exp(n^{1-1/40000000})$, with joint probability tending to one over the disorder and the sampled state.

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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.

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Cite

@misc{OAI:A-typical-start-upper-bound-for-low-temperature-SK-Glauber-dynamics-September-24-2026,
  author = {{OpenAI}},
  title = {{A typical-start upper bound for low-temperature SK Glauber dynamics}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-typical-start-upper-bound-for-low-temperature-SK-Glauber-dynamics-September-24-2026/paper.pdf}{OAI:A-typical-start-upper-bound-for-low-temperature-SK-Glauber-dynamics-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.

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