Result 227, Probability and statistical mechanics

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.

Lean formalization Proof

The bigger picture

Why it matters

In the Sherrington-Kirkpatrick model, n spins interact through random pairwise couplings without an external field. The unreviewed manuscripts claim a sharp temperature-dependent change in how quickly this system forgets its starting configuration.

What changes?

For Gaussian couplings and rate-one updates per spin, reported worst-start cutoff, an abrupt approach to equilibrium, occurs at log(n)/(2 lambda(beta)) for fixed inverse temperature 0 <= beta < 1, with deterministic lambda(beta) > 0. At beta = 1, worst-start mixing takes n^(2/3+o(1)). For fixed beta > 1, typical Gibbs-sampled configurations held fixed have bounds exp(n^(1/10000)) to exp(n^(1-1/40000000)). Claims hold with high probability over couplings and, for the upper bound, the sampled state.

What does that help mathematicians do?

At criticality, mean spin autocorrelations, which measure spin memory, are multiplied by n^(1/3), while waiting times and lags use units n^(2/3). Stationary and independent-fair-spin quench processes then reportedly have joint random limits unchanged between Gaussian and Rademacher, or random-sign, couplings. Convergence is uniform on compact sets of positive times. Researchers can thus distinguish universal memory behavior from persistent sample randomness. As waiting time grows, the quench limit approaches the stationary limit.

Are there practical applications?

The immediate value is foundational: these claims quantify equilibrium relaxation and its temperature-dependent barriers in a disordered model. They also inform the analysis of heat-bath sampling, rather than demonstrate a practical sampler. Crucially, the low-temperature upper bound concerns an equilibrium-sampled configuration subsequently held fixed, not an arbitrary initialization. Uniform single-spin update attempts require n times the rate-one-per-spin time scale.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

8 manuscripts

Universality of critical quench autocorrelations in the Sherrington–Kirkpatrick model

October 5, 2026 112 pages

We prove joint functional convergence of the stationary and quench autocorrelations of zero-field Sherrington–Kirkpatrick heat-bath dynamics at inverse temperature β = 1, with the same random limit for Gaussian and Rademacher couplings. Each site has a rate-one clock, mean spin autocorrelations are multiplied by n^1/3, and waiting times and lags are measured in units n^2/3. The quench starts from independent fair spins, and convergence is uniform on compact sets of positive waiting times and lags. The quench limit is selected by these initial states and relaxes to the stationary limiting autocorrelation as the waiting time tends to infinity.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Universality-of-critical-quench-autocorrelations-in-the-Sherrington-Kirkpatrick-model-October-5-2026,
  author = {{OpenAI}},
  title = {{Universality of critical quench autocorrelations in the Sherrington--Kirkpatrick model}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Universality-of-critical-quench-autocorrelations-in-the-Sherrington-Kirkpatrick-model-October-5-2026/main.pdf}{OAI:Universality-of-critical-quench-autocorrelations-in-the-Sherrington-Kirkpatrick-model-October-5-2026}},
  year = {2026}
}

Functional universality of critical SK autocorrelations

October 5, 2026 70 pages

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 n^2/3 and the mean spin autocorrelation is multiplied by n^1/3. 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.

PDF LaTeX source

Cite (BibTeX)

@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}
}

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

September 24, 2026 66 pages

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.

PDF LaTeX source

Cite (BibTeX)

@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}
}

Cutoff throughout the high-temperature Sherrington–Kirkpatrick phase

September 24, 2026 158 pages

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.

PDF LaTeX source

Cite (BibTeX)

@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}
}

Critical slowing down in the Sherrington–Kirkpatrick model

September 24, 2026 39 pages Main result formalized in Lean

At the critical inverse temperature β = 1, we prove slow mixing for zero-field Gaussian Sherrington–Kirkpatrick heat-bath dynamics from typical equilibrium configurations held fixed as initial states. For every deterministic sequence $t_n=o(n^{2/3})$ in rate-one-per-site time, the Gibbs mass of initial states whose time-t_n total-variation distance from equilibrium exceeds 1/4 tends to one in probability over the disorder. The same statement holds for every deterministic integer sequence $k_n=o(n^{5/3})$ of uniform-site update attempts.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026,
  author = {{OpenAI}},
  title = {{Critical slowing down in the Sherrington--Kirkpatrick model}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026/paper.pdf}{OAI:Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026}},
  year = {2026}
}

Stretched-exponential barriers for typical SK initial states

September 24, 2026 63 pages

At every fixed inverse temperature β > 1, we prove a stretched-exponential obstruction to mixing for the zero-field Sherrington–Kirkpatrick model from typical equilibrium configurations held fixed as initial states. The Gibbs mass of states whose total-variation distance from equilibrium at time $\exp(n^{1/10000})$ exceeds 1/4 tends to one in probability over the disorder. This holds for rate-one-per-site heat-bath dynamics and after the same stated number of discrete update attempts.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Stretched-exponential-barriers-for-typical-SK-initial-states-September-24-2026,
  author = {{OpenAI}},
  title = {{Stretched-exponential barriers for typical SK initial states}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Stretched-exponential-barriers-for-typical-SK-initial-states-September-24-2026/paper.pdf}{OAI:Stretched-exponential-barriers-for-typical-SK-initial-states-September-24-2026}},
  year = {2026}
}

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

September 24, 2026 22 pages

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.

PDF LaTeX source

Cite (BibTeX)

@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}
}

Critical mixing in the Sherrington–Kirkpatrick model

September 25, 2026 190 pages

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.

PDF LaTeX source

Cite (BibTeX)

@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's note on what the formalization covers, from lean/docs/227.md.

Critical SK autocorrelation processes and dynamics across the temperature transition

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

For every fixed inverse temperature $0<\beta<1$ , the formalization proves a dimension-independent Poincaré inequality for the zero-field Gaussian Sherrington–Kirkpatrick model with probability tending to one over the disorder. Equivalently, the unscaled single-site heat-bath spectral gap stays bounded away from zero. For dynamics that choose one site uniformly at each discrete step, the spectral gap is at least $1/(Cn)$ with probability tending to one, for a constant $C$ depending on $\beta$ .

The formalized supporting result proves ratio cutoff for discrete single-site heat-bath dynamics in the zero-field Gaussian Sherrington–Kirkpatrick model for $0\le\beta<1/2$ . For every $0<\varepsilon<1/2$ and $\eta>0$ , the probability over the disorder that $t_{\mathrm{mix}}(\varepsilon)/t_{\mathrm{mix}}(1-\varepsilon)>1+\eta$ tends to zero as $n\to\infty$ . The denominator is positive for all sufficiently large sizes.

The paper's full range $\beta<1$ , explicit cutoff location, and continuous-time conclusion are outside this selected statement.

At criticality in the zero-field Sherrington–Kirkpatrick model, the formalization proves that for every fixed $\varepsilon>0$ , the continuous-time mixing time lies between $n^{2/3-\varepsilon}$ and $e^{\varepsilon n}$ , and the discrete-time mixing count lies between $n^{5/3-\varepsilon}$ and $e^{\varepsilon n}$ , with probability tending to one over the Gaussian disorder. The disorder has independent off-diagonal entries of variance $1/n$ . Continuous updates have rate one per site; discrete time counts uniformly chosen single-site update attempts. Mixing is worst-start total variation at threshold $1/4$ , with holding allowed.

For deterministic times $t_n=o(n^{2/3})$ or update counts $k_n=o(n^{5/3})$ , the Gibbs mass of realized starting states still farther than $1/4$ from equilibrium tends to one in probability. For every deterministic positive sequence $M_n\to\infty$ , the covariance operator norm and the linear Rayleigh supremum, using the unscaled heat-bath Dirichlet form, both lie between $n^{2/3}/M_n$ and $M_n n^{2/3}$ with probability tending to one. The covariance result does not give an upper bound for the full inverse spectral gap.

The formalization proves a stretched-exponential mixing obstruction for the zero-field Sherrington–Kirkpatrick model at every fixed inverse temperature $\beta>1$ . At time $\exp(n^{1/10000})$ , the expected Gibbs mass of initial configurations whose total-variation distance from equilibrium exceeds $1/4$ tends to one. Since this mass lies in $[0,1]$ , it also tends to one in probability over the disorder.

The result covers both rate-one-per-site continuous-time heat-bath dynamics and the same stated number of discrete update attempts.

Comparator links

Result	Comparator statement
Spectral-gap bound throughout the high-temperature SK phase	SKHighTemperature.lean
Discrete mixing-time ratio cutoff for $\beta<1/2$	SKRatio.lean
Critical SK mixing bounds	CriticalSKMixing.lean
Equilibrium starts, covariance, and mixing lower bounds	CriticalSK.lean
Stretched-exponential barriers from typical equilibrium initial states	SKBarriers.lean

At a glance

Subject: Probability and statistical mechanics
Manuscripts: 8
Pages: 720
Dated: Sep 24 to Oct 5, 2026
Lean formalization: Yes
Reasoning summary: No

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