# Critical SK autocorrelation processes and dynamics across the temperature transition

> Heat-bath mixing in the SK spin glass is fast below β = 1, takes n^(2/3) at β = 1, and slows sharply above it.

Canonical: https://closedform.nishachay.tech/e/openai-math-2026-227/
Last updated: 2026-10-14

**Heat-bath mixing in the SK spin glass is fast below β = 1, takes n^(2/3) at β = 1, and slows sharply above it.**

**Trust:** Formal proof listed · **Papers:** 8 · **First paper:** September 24, 2026 · **Field:** Probability and statistical mechanics · **Entry:** 227

*Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error.

The Sherrington–Kirkpatrick model is a classic random magnet with n interacting spins. The papers study how fast a natural random update process reaches equilibrium. They claim fast mixing with a sharp cutoff at high temperature and a time of about n^(2/3) at the critical point. Below the critical temperature, typical starts take much longer.

## At a glance

- **First step:** 1975 (David Sherrington and Scott Kirkpatrick, the infinite-range spin glass)
- **Significance:** Major. It covers the whole high-temperature phase and fixes the critical mixing exponent for a central spin-glass model.
- **Trust:** Formal proof listed
- **Field:** Probability and statistical mechanics (8 papers)

## The explanation

Take n tiny magnets, called spins, each pointing up or down. Every pair is joined by a random strength, positive or negative. This is the Sherrington–Kirkpatrick (SK) model, introduced in 1975[1]. A number β, the inverse temperature, sets how strongly the spins feel those pulls. At β = 1 the model changes phase. Below 1 it behaves like a hot, loose system. Above 1 it becomes a spin glass, frozen in a rugged landscape.

Now let the spins move. Each spin carries a clock that rings on average once per time unit. When it rings, the spin is redrawn at random, given its neighbors. This is heat-bath dynamics. The mixing time is how long it takes, from the worst start, to look like a typical equilibrium sample.

The papers find three regimes. Take n = 1,000 spins. At β = 1, the mixing time grows like n^(2/3), here about 100 time units. That is about 100,000 single-spin updates, since each time unit has n of them. Below β = 1, mixing takes only about log n time units, here about 6.9 times a constant. Above β = 1, typical starts take time that grows faster than any power of n.

Earlier proofs covered only part of the hot phase. Bauerschmidt and Bodineau proved a key inequality for β below 1/4[2]. Others turned results like this into fast mixing in that range. Later work pushed the range up to 1/2, and slightly past it at zero field[3]. None of these bounds reach β = 1 by taking a limit[4]. Physicists had seen the n^(2/3) scale in simulations[5].

The spectral-gap paper proves that the dynamics stays well-mixed at every β below 1[6]. Equivalently, the equilibrium law satisfies a Poincaré inequality, a bound on how much any function can vary, that does not grow with n[7]. A second paper claims cutoff for every β below 1[8]. Cutoff means the distance from equilibrium stays near its maximum and then drops suddenly, at time log n / (2λ(β)) for an explicit rate λ(β).

At β = 1, one paper proves that typical equilibrium starts are still far from equilibrium before time n^(2/3)[9]. The critical mixing paper claims the matching upper bound, so the mixing time is n^(2/3 + o(1))[10]. The o(1) means the exponent is right up to any small slack. It does not claim a cutoff or a leading constant[11]. The obstacle at β = 1 is that equilibrium fluctuations are much larger than for independent spins[12]. Every bound must hold from every start and for every test function.

Above β = 1, one paper proves a barrier for typical starts. Starting from a sampled equilibrium state, the dynamics is still far from equilibrium at time exp(n^(1/10000))[13]. That bound only matters for huge n. For a million spins it is about 2.72 time units. Another paper claims an upper bound for typical starts, below exp(n) by a tiny power[14].

Two more papers look at β = 1 more closely. They scale time by n^(2/3) and multiply the average spin autocorrelation by n^(1/3)[15]. With n = 1,000 that factor is 10. The limit is the same for Gaussian and ±1 strengths[16]. It stays random from sample to sample[17]. One paper starts from independent fair spins, a sudden cooling called a quench[18]. Its limit relaxes to the equilibrium one over long waiting times[19].

How sure is this? Lean, a computer proof checker, has formal versions of several pieces. They include the spectral gap for every β below 1, and cutoff for β below 1/2. At β = 1, Lean bounds the mixing time between n^(2/3−ε) and e^(εn), so the matching upper bound is not in it. It also covers the β > 1 barrier. The universality papers and the β > 1 upper bound are not covered. No specialist review of these papers has been published.

**What it doesn't show.** This does not give a cutoff or a leading constant at the critical point.

## Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

## Hype check

Major if the proofs hold. Lean covers the spectral gap, cutoff below β = 1/2, critical lower bounds and the β > 1 barrier. The critical upper exponent and universality are not covered, and no specialist review is published.

## Words used

- **SK model:** n spins with independent random pairwise strengths, all pairs interacting.
- **Inverse temperature β:** A setting that scales how strongly spins feel their interactions.
- **Heat-bath dynamics:** A process that repeatedly redraws one spin from its correct conditional law given the others.
- **Mixing time:** How long the process needs, from the worst start, to be close to equilibrium.
- **Spectral gap:** A rate that controls how fast averages settle; bounded away from zero means fast settling.
- **Cutoff:** A sudden drop from far to near equilibrium within a short time window.
- **Autocorrelation:** How much a spin still agrees with its earlier value after a time lag.

## The original papers (8)

- [Universality of critical quench autocorrelations in the Sherrington–Kirkpatrick model](https://github.com/openai/math/blob/main/preprints/Universality-of-critical-quench-autocorrelations-in-the-Sherrington-Kirkpatrick-model-October-5-2026/main.pdf) - October 5, 2026
- [Functional universality of critical SK autocorrelations](https://github.com/openai/math/blob/main/preprints/Functional-universality-of-critical-SK-autocorrelations-October-5-2026/critical-sk-autocorrelations.pdf) - October 5, 2026
- [A spectral gap throughout the high-temperature Sherrington–Kirkpatrick phase](https://github.com/openai/math/blob/main/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/main.pdf) - September 24, 2026
- [Cutoff throughout the high-temperature Sherrington–Kirkpatrick phase](https://github.com/openai/math/blob/main/preprints/Cutoff-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/paper.pdf) - September 24, 2026
- [Critical slowing down in the Sherrington–Kirkpatrick model](https://github.com/openai/math/blob/main/preprints/Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026/paper.pdf) - September 24, 2026 · Formal proof listed
- [Stretched-exponential barriers for typical SK initial states](https://github.com/openai/math/blob/main/preprints/Stretched-exponential-barriers-for-typical-SK-initial-states-September-24-2026/paper.pdf) - September 24, 2026
- [A typical-start upper bound for low-temperature SK Glauber dynamics](https://github.com/openai/math/blob/main/preprints/A-typical-start-upper-bound-for-low-temperature-SK-Glauber-dynamics-September-24-2026/paper.pdf) - September 24, 2026
- [Critical mixing in the Sherrington–Kirkpatrick model](https://github.com/openai/math/blob/main/preprints/Critical-mixing-in-the-Sherrington-Kirkpatrick-model-September-25-2026/paper.pdf) - September 25, 2026

- [Lean notes](https://github.com/openai/math/blob/main/lean/docs/227.md)

## Quoted lines from the paper (19)

[1] From [Critical mixing in the Sherrington–Kirkpatrick model](https://github.com/openai/math/blob/main/preprints/Critical-mixing-in-the-Sherrington-Kirkpatrick-model-September-25-2026/paper.pdf):

> Sherrington and Kirkpatrick introduced the infinite-range spin-glass model in 1975.

[2] From [Critical mixing in the Sherrington–Kirkpatrick model](https://github.com/openai/math/blob/main/preprints/Critical-mixing-in-the-Sherrington-Kirkpatrick-model-September-25-2026/paper.pdf):

> Bauerschmidt and Bodineau proved a logarithmic Sobolev inequality under a spectral-width condition that applies to SK for fixed $\beta<1/4$.

[3] From [Critical mixing in the Sherrington–Kirkpatrick model](https://github.com/openai/math/blob/main/preprints/Critical-mixing-in-the-Sherrington-Kirkpatrick-model-September-25-2026/paper.pdf):

> At zero field, Boban, Li, and Oveis Gharan obtained a constant per-site spectral gap and an $O(n^2)$ attempted-update bound for $\beta<1/2+\varepsilon_0$, with a universal $\varepsilon_0>0$.

[4] From [Critical mixing in the Sherrington–Kirkpatrick model](https://github.com/openai/math/blob/main/preprints/Critical-mixing-in-the-Sherrington-Kirkpatrick-model-September-25-2026/paper.pdf):

> None of these fixed-subcritical bounds supplies a bound at $\beta=1$ by taking a limit.

[5] From [Critical mixing in the Sherrington–Kirkpatrick model](https://github.com/openai/math/blob/main/preprints/Critical-mixing-in-the-Sherrington-Kirkpatrick-model-September-25-2026/paper.pdf):

> For heat-bath updates, with time measured in updates per spin, it tested the finite-size scaling form $q_c(n,t)\sim n^{-1/3}F(t/n^{2/3})$.

[6] From [A spectral gap throughout the high-temperature Sherrington–Kirkpatrick phase](https://github.com/openai/math/blob/main/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/main.pdf):

> For every fixed inverse temperature $0<\beta<1$, 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.

[7] From [A spectral gap throughout the high-temperature Sherrington–Kirkpatrick phase](https://github.com/openai/math/blob/main/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/main.pdf):

> Equivalently, the Gibbs law satisfies a dimension-free Poincar\'e inequality for all functions.

[8] From [Cutoff throughout the high-temperature Sherrington–Kirkpatrick phase](https://github.com/openai/math/blob/main/preprints/Cutoff-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/paper.pdf):

> We prove worst-case total-variation cutoff for the zero-field Gaussian Sherrington--Kirkpatrick heat-bath dynamics at every fixed inverse temperature \(0\leq\beta<1\).

[9] From [Critical slowing down in the Sherrington–Kirkpatrick model](https://github.com/openai/math/blob/main/preprints/Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026/paper.pdf):

> At the critical inverse temperature $\beta=1$, we prove slow mixing for zero-field Gaussian Sherrington--Kirkpatrick heat-bath dynamics from typical equilibrium configurations held fixed as initial states.

[10] From [Critical mixing in the Sherrington–Kirkpatrick model](https://github.com/openai/math/blob/main/preprints/Critical-mixing-in-the-Sherrington-Kirkpatrick-model-September-25-2026/paper.pdf):

> At the critical inverse temperature $\beta=1$, we determine the worst-start total-variation mixing exponent for single-site heat-bath dynamics in the zero-field Gaussian Sherrington--Kirkpatrick model.

[11] From [Critical mixing in the Sherrington–Kirkpatrick model](https://github.com/openai/math/blob/main/preprints/Critical-mixing-in-the-Sherrington-Kirkpatrick-model-September-25-2026/paper.pdf):

> It gives the mixing exponent and does not assert a cutoff, a cutoff window, or a leading constant.

[12] From [Critical mixing in the Sherrington–Kirkpatrick model](https://github.com/openai/math/blob/main/preprints/Critical-mixing-in-the-Sherrington-Kirkpatrick-model-September-25-2026/paper.pdf):

> At criticality, equilibrium fluctuations are much larger than for independent spins.

[13] From [Stretched-exponential barriers for typical SK initial states](https://github.com/openai/math/blob/main/preprints/Stretched-exponential-barriers-for-typical-SK-initial-states-September-24-2026/paper.pdf):

> At every fixed inverse temperature $\beta>1$, we prove a stretched-exponential obstruction to mixing for the zero-field Sherrington--Kirkpatrick model from typical equilibrium configurations held fixed as initial states.

[14] From [A typical-start upper bound for low-temperature SK Glauber dynamics](https://github.com/openai/math/blob/main/preprints/A-typical-start-upper-bound-for-low-temperature-SK-Glauber-dynamics-September-24-2026/paper.pdf):

> For every fixed inverse temperature $\beta>1$, 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.

[15] From [Functional universality of critical SK autocorrelations](https://github.com/openai/math/blob/main/preprints/Functional-universality-of-critical-SK-autocorrelations-October-5-2026/critical-sk-autocorrelations.pdf):

> With rate-one heat-bath clocks at every site, time is scaled by $n^{2/3}$ and the mean spin autocorrelation is multiplied by $n^{1/3}$.

[16] From [Functional universality of critical SK autocorrelations](https://github.com/openai/math/blob/main/preprints/Functional-universality-of-critical-SK-autocorrelations-October-5-2026/critical-sk-autocorrelations.pdf):

> We prove that the stationary spin autocorrelation of the zero-field Sherrington--Kirkpatrick model at inverse temperature $\beta=1$ has a common functional scaling limit for Gaussian and Rademacher couplings.

[17] From [Functional universality of critical SK autocorrelations](https://github.com/openai/math/blob/main/preprints/Functional-universality-of-critical-SK-autocorrelations-October-5-2026/critical-sk-autocorrelations.pdf):

> Their limiting law is not a point mass: it retains sample-to-sample randomness, and its functions decay to zero at large times.

[18] From [Universality of critical quench autocorrelations in the Sherrington–Kirkpatrick model](https://github.com/openai/math/blob/main/preprints/Universality-of-critical-quench-autocorrelations-in-the-Sherrington-Kirkpatrick-model-October-5-2026/main.pdf):

> The quench starts from independent fair spins, and convergence is uniform on compact sets of positive waiting times and lags.

[19] From [Universality of critical quench autocorrelations in the Sherrington–Kirkpatrick model](https://github.com/openai/math/blob/main/preprints/Universality-of-critical-quench-autocorrelations-in-the-Sherrington-Kirkpatrick-model-October-5-2026/main.pdf):

> The quench limit is selected by these initial states and relaxes to the stationary limiting autocorrelation as the waiting time tends to infinity.

## Other sources

- [openai/math release notes (2026): model provenance and review status](https://github.com/openai/math/blob/main/README.md): “manuscripts and supporting proof artifacts produced by an internal OpenAI model”

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