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Cutoff throughout the high-temperature Sherrington–Kirkpatrick phase

OpenAI · September 24, 2026

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Abstract

We prove worst-case total-variation cutoff for the zero-field Gaussian Sherrington–Kirkpatrick heat-bath dynamics at every fixed inverse temperature $0\leq\beta\lt 1$. With rate-one refresh at each spin, the cutoff location is $\log n/(2\lambda(\beta))$ for a positive deterministic rate $\lambda(\beta)$. The location for uniformly chosen single-site update attempts is n times as large. Convergence is 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 (lean/docs/227.md, 2026-10-06). The formalized statement is ratio cutoff, t_mix(ε)/t_mix(1−ε) → 1 in probability, for discrete heat-bath dynamics at $0\le\beta<1/2$ (comparator SKRatio.lean); the full range β < 1, the explicit cutoff location and the continuous-time statement are not formalized. [family-level]

Possibly relevant literature (found by the agent, not checked)

Digests and expositions

Elsewhere

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Cite

@misc{OAI:Cutoff-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026,
  author = {{OpenAI}},
  title = {{Cutoff throughout the high-temperature Sherrington--Kirkpatrick phase}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Cutoff-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/paper.pdf}{OAI:Cutoff-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-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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