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A spectral gap throughout the high-temperature Sherrington–Kirkpatrick phase

OpenAI · September 24, 2026

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Abstract

For every fixed inverse temperature $0\lt \beta\lt 1$, we prove that the unscaled spectral gap of single-site heat-bath dynamics for the zero-field Gaussian Sherrington–Kirkpatrick model is bounded away from zero with probability tending to one over the disorder. Equivalently, the Gibbs law satisfies a dimension-free Poincaré inequality for all functions.

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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). States that the release's Lean development proves the dimension-free zero-field Gaussian SK Poincaré inequality of Theorem 1.1 for every fixed $0<\beta<1$ with probability tending to one, with uniform-site discrete gap at least $1/(C_\beta n)$ (comparator SKHighTemperature.lean). [family-level]

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

Digests and expositions

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Cite

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

Other papers in this family: