SK dynamics across the temperature transition

Source and verification

This article explains family 227 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

The eight papers in family 227 describe heat-bath dynamics for the Sherrington–Kirkpatrick model across its temperature transition. They distinguish equilibrium fluctuations, worst-start mixing, mixing from a typical fixed configuration, and relaxation after a quench.

🏷️ Preliminaries: heat-bath dynamics and its clocks

For independent standard Gaussian couplings $J_{ij}$, define the zero-field Gibbs law on $n$ Ising spins by

$${\pi }_{n,\beta }^J(\sigma )\propto \exp \left(\frac{\beta }{\sqrt{n}}\sum _{i<j}{J}_{ij}{\sigma }_i{\sigma }_j\right).$$

At rate one, each site resamples its spin from its conditional Gibbs law. This is a reversible Markov chain with stationary measure ${\pi }_{n,\beta }^J$. One unit of continuous time contains $n$ attempted updates on average. A discrete chain choosing one uniform site per step therefore uses a clock larger by a factor of $n$.

For a fixed initial configuration $x$, total variation measures the discrepancy between the transition law $P_t^J(x,\cdot )$ and equilibrium. The worst-start mixing time takes the maximum over $x$. The spectral gap controls decay in the equilibrium $L^2$ norm through the Poincaré inequality

$$\begin{array}{c}{\mathrm{Var}}_{\pi }(f)\le {\mathrm{gap}}^{-1}\mathcal{E}(f,f),\\ \mathcal{E}(f,f)=-⟨f,Lf{⟩}_{\pi }.\end{array}$$

A positive gap alone does not specify the sharp worst-start mixing location.

🏷️ The temperature regimes

At each fixed $0<\beta <1$, the gap companion claims a positive dimension-independent lower bound with probability tending to one over the disorder. (OpenAI, 2026) The cutoff companion treats $0\le \beta <1$ and claims worst-start total-variation cutoff at

$$t_n=\frac{\log n}{2\lambda (\beta )},\lambda (\beta )>0.$$

Here cutoff means an abrupt asymptotic transition around this location on its relative scale. The update-attempt location is $n{t}_n$. The rate is defined through the manuscript’s equilibrium and gradient analysis; it is not replaced here by a guessed elementary function of $\beta$. (OpenAI, 2026b)

At $\beta =1$, the critical mixing companion claims worst-start mixing time $n^{2/3+o(1)}$, or $n^{5/3+o(1)}$ attempted updates, in disorder probability. (OpenAI, 2026b) Its slowing-down companion proves a more specific obstruction: at every deterministic $t_n=o(n^{2/3})$, the Gibbs mass of fixed starting configurations still at distance greater than $1/4$ tends to one. (OpenAI, 2026c)

At each fixed $\beta >1$, a barrier companion claims the same typical-start obstruction at time $\exp (n^{1/10000})$. (OpenAI, 2026e) A different companion claims a typical-start upper bound $\exp (n^{1-c_{\beta }})$, with $c_{\beta }>0$; its proof permits $c_{\beta }=1/40000000$. These bounds leave a substantial gap and do not identify an exact low-temperature mixing exponent. (OpenAI, 2026b)

A configuration sampled from equilibrium is realized and then held fixed before its transition law is tested. Averaging that starting configuration into the transition law would make the law stationary at every time and erase the question.

🏷️ Independent-spin calibration

At $\beta =0$, start with every spin equal to $+1$. Each spin retains mean $e^{-t}$, because its initial value survives precisely until its first refresh. The magnetization therefore has mean $n{e}^{-t}$, while its equilibrium standard deviation is $\sqrt{n}$. The two scales meet at $t=\frac{1}{2}\log n$.

This elementary calculation explains the logarithmic location and the factor $1/2$. For interacting spins the proof needs an appropriate slowly decaying observable for the lower bound and uniform control of all bounded tests for the upper bound. The cutoff paper propagates exact discrete gradients under heat flow and combines that propagation with a late-time energy estimate.

🏷️ Critical autocorrelation processes

The stationary mean spin autocorrelation is

$$C_{n,J}(t)=\frac{1}{n}\sum _i⟨{\sigma }_i,P_t^J{\sigma }_i{⟩}_{{\pi }_{n,1}^J}.$$

The functional companion claims that $n^{1/3}{C}_{n,J}(s{n}^{2/3})$, for $s>0$, converges in law uniformly on compact positive-time intervals. The same nonconstant random limiting law occurs for Gaussian and uniform-sign couplings. Thermal and dynamical randomness have already been averaged; the surviving randomness is that of the couplings. Time zero is excluded, since the rescaled value there equals $n^{1/3}$. (OpenAI, 2026f)

The quench companion starts instead from independent fair spins, waits $s{n}^{2/3}$, then measures a lag $t{n}^{2/3}$. It claims joint functional convergence of stationary and quench correlations, uniformly on compact sets with $s,t>0$. The quench limit relaxes to the stationary limit as its waiting time tends to infinity. This is an iterated limiting statement, not a uniform assertion down to waiting time zero. (OpenAI, 2026h)

🏷️ Structure of the dynamical arguments

For high temperature, observing spins in Gaussian noise produces posterior Gibbs laws with smaller conditional uncertainty. A terminal Poincaré estimate must be transported back along this observation process, controlling exceptional paths and covariance amplification.

At criticality, eigenmodes near the upper edge of the completed interaction matrix have the relevant $n^{-2/3}$ spectral scale. Static convergence of these coordinates does not determine their dynamics: microscopic corrections can lower Dirichlet energy, and finitely many modes do not control the full autocorrelation sum. The functional paper proves relaxed-energy and resolvent-trace estimates before recovering time correlations. The quench extension additionally regularizes the nonstationary density reached after positive waiting time.

Below the temperature transition, the lower bound constructs an overlap barrier relative to an equilibrium reference configuration. Spin reversal gives a large target region on the opposite side, which a mixed chain would need to reach. The upper bound instead constructs proposals covering sets of appreciable Gibbs mass and joins proposals by controlled small Hamming moves. It rules out cuts with too little stationary flow for typical starts. These are different mechanisms, explaining why neither result supplies a sharp common timescale.

🏷️ Key steps and proof outlines

Transporting a posterior gap back to the original system

The high-temperature gap argument observes an equilibrium spin $X$ in Gaussian noise. If $Y(t)$ is the observation and ${\mathbb{E}}_t$ denotes its posterior law, the innovation process is Brownian and Bayes’ formula gives

$$d({\mathbb{E}}_{t}f)={\mathrm{Cov}}_t(f,X{)}^{T}d{I}_t.$$

For a centered test with initial variance one, let $V_t={\mathbb{E}}_t{f}^2-({\mathbb{E}}_{t}f{)}^2$ and $d_0={\mathcal{E}}_0(f,f)$. Stop at a fixed horizon or before posterior stability fails, using a rule independent of $f$. The covariance estimate and conditional-variance contraction of the posterior Dirichlet form give

$$u^{\prime }(t)\ge -C_{\mathrm{d}}(u(t)+d_0),u(t)=\mathbb{E}{V}_{t\wedge \tau }.$$

Hence $\mathbb{E}{V}_{\tau }\ge v-(1-v)d_0$, where $v=e^{-C_{\mathrm{d}}T}>0$. On good paths the terminal Poincaré inequality gives an upper bound $2d_0$.

It suffices to consider $d_0\le 1$, since larger energy already implies the final lower bound. A small ordinary probability of bad paths is insufficient: $f^2$ could concentrate there. The source also controls the observation law tilted by $f^2$. Its stopped-density identity gives $\mathbb{E}[1_{\mathrm{bad}}{V}_{\tau }]\le v/4$ uniformly over normalized tests. Combining the bounds yields

$$v-(1-v)d_0\le 2d_0+v/4,d_0\ge \frac{3v}{4(3-v)}\ge v/4.$$

This uniform lower bound on energy at unit variance is the claimed spectral gap. The weighted exceptional-path estimate is what makes the argument apply to every test function simultaneously. (OpenAI, 2026a)

The extra estimate needed for cutoff

For a cube function $g$, let $d_{i}g$ be its half-difference in coordinate $i$. Its multilinear extension to $[-1,1{]}^n$ satisfies ${\partial }_i\bar{g}={\mathbb{E}}_z{d}_{i}g$. Jensen’s inequality and integration along a segment of length at most $2\sqrt{n}$ prove

$$\mathrm{osc}(g)\le 2\sqrt{n}\underset{x}{\sup }\Vert dg(x){\Vert }_2.$$

Thus the cutoff upper bound needs the gradient of $P_{T}f$ to be $o(n^{-1/2})$, uniformly over $|f|\le 1$ and starting states. A mere equilibrium variance bound is not enough.

The backward martingale $P_{T-s}f(X_s)$ has total expected bracket at most one. This gives an integrated weighted-gradient budget on a late interval. Field-tail estimates keep the heat-bath weights at least $n^{-2\delta }$ except on a negligible event, so some time in that interval has a controlled unweighted gradient energy. The exact gradient propagation estimate transports this energy back to time zero at rate arbitrarily close to $\lambda (\beta )$. For $T>(1+\epsilon )\log n/(2\lambda )$ it beats $n^{-1/2}$; the oscillation estimate then controls all bounded tests and proves total-variation convergence.

For the lower bound, the manuscript constructs a starting state with a macroscopic mean in a visible slow direction and controls its fluctuations. When the rate comes from an isolated visible eigenspace, the preparation and alignment argument is necessary; selecting an arbitrary top matrix eigenvector would not establish the required slow mean. (OpenAI, 2026e)

A critical slow coordinate distinguishes typical fixed starts

For any reversible chain,

$${\mathbb{E}}_{\pi }[(f(X_t)-f(X_0){)}^2]=2⟨f,(1-P_t)f{⟩}_{\pi }\le 2t\mathcal{E}(f,f).$$

This follows from $1-e^{-t\lambda }\le t\lambda$ in the spectral decomposition. For $f(x)=v^{T}x$ with $\Vert v{\Vert }_2=1$, the heat-bath form is at most one. The critical static theorem supplies a top-edge direction whose equilibrium projection has scale $n^{1/3}$ and little mass in smaller central windows.

For $t_n=o(n^{2/3})$, choose $a_n=b_n{n}^{1/3}$ with $b_n\downarrow 0$ but $t_n/a_n^2\to 0$. The probability of a sign change is bounded by

$$\pi \{|v^T{X}_0|\le a_n\}+\frac{2t_n}{a_n^2}=o(1).$$

For each realized start $x$, test its transition law against the half-space $H_{v,x}=\{y:(v^{T}y)(v^{T}x)>0\}$. Spin reversal makes its equilibrium mass at most $1/2$, while the evolving chain stays there with high probability for most starts. This gives total variation at least $1/2-o(1)$ in the required averaged fixed-start sense.

The eigenvector may use an auxiliary Gaussian diagonal added to the interaction matrix. That diagonal changes neither the Gibbs law nor the dynamics. The proof first bounds the intrinsic indicator that a fixed state has already mixed, and only then averages out the diagonal. It never averages the transition law over equilibrium starts before testing it. (OpenAI, 2026d)

A support-sensitive energy bound supplies the matching critical upper exponent

The critical-mixing companion proves, on one good-disorder event, a functional inequality of the form

$$\mathcal{E}(f,f)\ge \alpha \pi (f^2)\log (1/p),p=\pi (f>0)\le 1/2,$$

for every nonnegative $f$, with $\alpha =n^{-2/3-o(1)}$. To see how it controls worst starts, let $g$ be a probability density and $V=\pi (g^2)$. Truncate it at $V/4$: $u=(g-V/4{)}_{+}$. Then

$$\pi (u>0)\le 4/V,\pi (u^2)\ge V/2,\mathcal{E}(g,g)\ge \mathcal{E}(u,u).$$

For $V\ge 8$ this yields $\mathcal{E}(g,g)\ge (\alpha /2)V\log (V/4)$. Along heat flow, $V^{\prime }=-2\mathcal{E}(g,g)$, so $z=\log (V/4)$ satisfies $z^{\prime }\le -\alpha z$. A point mass starts with $V=1/\pi (x)$, whose logarithm is $O(n)$ on the matrix event. Thus $V$ reaches a bounded value in $O({\alpha }^{-1}\log n)$ time. A Poincaré bound obtained by splitting a test at its median handles the final decay. The logarithmic factor is absorbed in $n^{o(1)}$, giving the upper exponent $2/3$. (OpenAI, 2026c)

Low-temperature barriers and balanced flows answer different questions

For the lower bound, choose a reference spin $v$ from an independent bank of equilibrium samples, orienting it to have positive overlap with the initial state. The selection uses no future updates. If the chain had mixed, spin reversal would force at least probability $1/4$ of reaching nonpositive overlap with this reference. But one update changes overlap by at most $2/n$.

When the scalar analysis supplies a forbidden overlap interval, its disorder-averaged equilibrium pair probability is at most $e^{-cn}$. For each fixed bank entry and deterministic time, stationarity and conditional independence permit this estimate after averaging over disorder. A union bound over the entire raw bank and all update indices controls them simultaneously, even though the selected reference depends on the start. Any passage from positive to nonpositive overlap must then hit the forbidden interval, giving a contradiction. The other scalar regime lacks such a fixed interval; it requires the manuscript’s narrow-locking estimates, additional banks, and conflict deletion. The simple forbidden-interval argument alone does not cover the whole low-temperature theorem. (OpenAI, 2026g)

For the upper bound, the source proves flow across every cut whose two sides have appreciable mass. Fix $0<\epsilon <1/3$. A finite pruning argument removes subsets with poor internal expansion. If the removed mass first reached $\epsilon$, its union would be a balanced cut with too little flow, contradicting the hypothesis. Thus a set of mass greater than $1-\epsilon$ remains with internal expansion.

Coarea and Cauchy–Schwarz give a Poincaré inequality on this set. Apply it to square roots of transition densities, and integrate the full chain’s entropy dissipation. If total jump rate is at most $R$ and balanced cuts have flow greater than $B$, the resulting bound is

$$\sum _x\pi (x)\Vert P_T(x,\cdot )-\pi {\Vert }_{\mathrm{TV}}\le \frac{\sqrt{2RH(\pi )}}{B\sqrt{T}}+2\epsilon .$$

For SK, $R=n$ and $H(\pi )\le n\log 2$. The constructed subexponential $B^{-1}$ permits the stated upper time. No trajectory is required to stay inside the retained set. Markov’s inequality converts the average of fixed-state distances into the typical-start conclusion. (OpenAI, 2026b)

Autocorrelations require control of infinitely many edge modes

The functional companion first identifies the limiting relaxed Dirichlet form on each fixed collection of edge coordinates. Summing those coordinates requires another estimate. With $H=-L$, for $z>0$ a source $S$ has the variational resolvent formula

$$⟨S,(z+n^{2/3}H{)}^{-1}S⟩=\underset{f}{\sup }\{2⟨S,f⟩-z\Vert f{\Vert }_2^2-n^{2/3}\mathcal{E}(f,f)\}.$$

Angular integration by parts uses the $a$th edge gap, of order $a^{2/3}$, to gain $a^{-2/3}$ in the dual source norm. Squaring gives the summable tail $a^{-4/3}$. This controls the resolvent trace uniformly beyond a growing number of coordinates and permits passage from finite-mode convergence to the full autocorrelation. Separate disorder replacement estimates then transfer the Gaussian limit to uniform-sign couplings. (OpenAI, 2026f)

The quench companion additionally selects an entrance law at positive waiting times. Its densities $h_s$ satisfy $h_{s+t}=S_t{h}_s$ and are locally bounded for $s>0$. The retained initial information is an escape condition: along times decreasing to zero, each fixed normalized coordinate $x_i^2/M(x)$ vanishes in probability under $h_s$, where $M(x)={\sum }_i{x}_i^2/(2r_i)$ and $r_i\asymp i^{2/3}$. The source’s angular realization and renormalized Volterra equation couple candidate entrances and prove uniqueness from this condition. It then derives the zero-coordinate limit, rather than assuming it determines an infinite-dimensional entrance. Warming estimates and clock-score replacement identify the same entrance for the two disorder laws. Positive waiting time is essential to these bounds. (OpenAI, 2026h)

See Also

• Family 217 concerns fluctuations of the static log partition function rather than dynamical relaxation.
• Family 238 provides a second example where mixing the full law requires more than controlling individual observables.

🏷️ Manuscripts and verification scope

• Universality of critical quench autocorrelations in the Sherrington–Kirkpatrick model — pinned manuscript and companion source.
• Functional universality of critical SK autocorrelations — pinned manuscript and companion source.
• A spectral gap throughout the high-temperature Sherrington–Kirkpatrick phase — pinned manuscript and companion source.
• Cutoff throughout the high-temperature Sherrington–Kirkpatrick phase — pinned manuscript and companion source.
• Critical slowing down in the Sherrington–Kirkpatrick model — pinned manuscript and companion source.
• Stretched-exponential barriers for typical SK initial states — pinned manuscript and companion source.
• A typical-start upper bound for low-temperature SK Glauber dynamics — pinned manuscript and companion source.
• Critical mixing in the Sherrington–Kirkpatrick model — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The selected high-temperature gap statement covers fixed beta below one. The cutoff selection is only discrete ratio cutoff below beta one-half, not the full location theorem below one. The critical selection gives a polynomial lower bound and subexponential upper bound, not the later sharp mixing exponent. It also documents typical-start critical and low-temperature obstructions. The functional autocorrelation, quench, and low-temperature upper-bound companions are not included in these selections. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. A spectral gap throughout the high-temperature Sherrington–Kirkpatrick phase.

🐻  OpenAI 2026b. A typical-start upper bound for low-temperature SK Glauber dynamics.

🐻  OpenAI 2026c. Critical mixing in the Sherrington–Kirkpatrick model.

🐻  OpenAI 2026d. Critical slowing down in the Sherrington–Kirkpatrick model.

🐻  OpenAI 2026e. Cutoff throughout the high-temperature Sherrington–Kirkpatrick phase.

🐻  OpenAI 2026f. Functional universality of critical SK autocorrelations.

🐻  OpenAI 2026g. Stretched-exponential barriers for typical SK initial states.

🐻  OpenAI 2026h. Universality of critical quench autocorrelations in the Sherrington–Kirkpatrick model.