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Functional universality of critical SK autocorrelations

OpenAI · October 5, 2026

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Abstract

We prove that the stationary spin autocorrelation of the zero-field Sherrington–Kirkpatrick model at inverse temperature β = 1 has a common functional scaling limit for Gaussian and Rademacher couplings. With rate-one heat-bath clocks at every site, time is scaled by n2/3 and the mean spin autocorrelation is multiplied by n1/3. The functions converge in law uniformly on compact positive-time intervals. Their limiting law is not a point mass: it retains sample-to-sample randomness, and its functions decay to zero at large times.

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

  • Yan Ru Pei, Disorder transfer for the critical SK overlap law (arXiv:2609.14273, 2026-09-13). Its conditional transfer of the Gaussian critical equilibrium overlap limit to bounded symmetric couplings, including fair signs, may inform the static disorder-comparison component of this paper’s dynamical universality claim. [family-level]

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Cite

@misc{OAI:Functional-universality-of-critical-SK-autocorrelations-October-5-2026,
  author = {{OpenAI}},
  title = {{Functional universality of critical SK autocorrelations}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Functional-universality-of-critical-SK-autocorrelations-October-5-2026/critical-sk-autocorrelations.pdf}{OAI:Functional-universality-of-critical-SK-autocorrelations-October-5-2026}},
  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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