The dynamical phase transition in the Sherrington–Kirkpatrick model

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Family 227 · Probability and statistical mechanics
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-08

A system can have a perfectly well-defined equilibrium distribution and still take an enormous time to reach it. Family 227 studies this gap in a central model of disordered magnetism. Its eight manuscripts claim a transition from logarithmic relaxation time to a critical power law and then to much slower relaxation. Understanding the claim requires care about both the clock and the initial state.

1. The question

Imagine many tiny magnets, each pointing either up or down. Some pairs prefer to agree; others prefer to disagree. These preferences can conflict: three magnets cannot all disagree with each other. Such frustration produces competing configurations rather than a single obvious arrangement.

Allow one magnet at a time to reconsider its direction, with some thermal randomness. How long until the distribution of configurations forgets where it started?

This is a question about mixing time. Knowing which configurations are likely at equilibrium does not tell us how quickly local updates explore them. Family 227 asks how that exploration changes as temperature crosses the spin-glass transition. The precise results discussed below are claims of the linked manuscripts; their scope and limitations are discussed in Section 7.

2. Background and essential definitions

Spins and random interactions

A configuration is a vector $\sigma=(\sigma_1,\ldots,\sigma_n)\in\{-1,+1\}^n$. For each pair $i<j$, draw an independent standard Gaussian variable $J_{ij}$, meaning a normal variable of mean zero and variance one. These couplings are drawn once and held fixed. They are called the disorder.

Use the interaction score

$$H_J(\sigma)=\frac{1}{\sqrt n}\sum_{i<j}J_{ij}\sigma_i\sigma_j.$$

Positive couplings reward agreement; negative couplings reward disagreement. Every pair interacts, so this is a mean-field model. The normalization keeps the typical total field acting on one spin of order one. With this sign convention, physical energy is $-H_J$.

At inverse temperature $\beta$, the equilibrium, or Gibbs, probability is

$$\pi_{J,\beta}(\sigma)=\frac{e^{\beta H_J(\sigma)}}{Z_{J,\beta}}, \qquad Z_{J,\beta}=\sum_{\tau\in\{-1,+1\}^n}e^{\beta H_J(\tau)}.$$

The partition function $Z_{J,\beta}$ makes the probabilities sum to one. Larger $\beta$ means lower temperature and stronger preference for favorable interactions. Zero external field means no additional term favors up or down spins individually. In this normalization, the critical inverse temperature is $\beta=1$.

The dynamics and its clock

In heat-bath Glauber dynamics, each spin has an independent clock ringing at rate one. At a ring, redraw that spin from its conditional equilibrium distribution, keeping the others fixed. Writing $\sigma_{-i}$ for all spins except spin $i$, the rule is

$$\begin{aligned} h_i(\sigma)&=\frac{1}{\sqrt n}\sum_{j\ne i}J_{ij}\sigma_j,\\ \mathbb P(\sigma_i\leftarrow s\mid\sigma_{-i}) &=\frac{e^{\beta s h_i(\sigma)}}{2\cosh(\beta h_i(\sigma))}. \end{aligned}$$

Here $s\in\{-1,+1\}$, $J_{ji}=J_{ij}$, and $2\cosh u=e^u+e^{-u}$. A redraw may leave the spin unchanged. The future depends only on the present configuration, making this a Markov chain. Its stationary distribution is $\pi_{J,\beta}$.

One unit of time means approximately one attempted update per spin: $n$ attempts altogether. Counting individual random-site attempts instead multiplies the relevant time scales by $n$. Confusing these clocks would change the critical exponent from $2/3$ to $5/3$.

Measuring the memory of an initial state

For distributions $\mu,\nu$ on the configurations, their total-variation distance is

$$\|\mu-\nu\|_{\mathrm{TV}} =\frac12\sum_\sigma|\mu(\sigma)-\nu(\sigma)|.$$

It is also the largest difference between their probabilities of the same event. Distance near zero means that no event reliably distinguishes the distributions.

Let $P_t(x,\cdot)$ be the law at time $t$ starting from configuration $x$. Suppressing $J,\beta$ in the notation, define

$$\begin{aligned} T(x)&=\inf\{t\ge0:\|P_t(x,\cdot)-\pi\|_{\mathrm{TV}}\le1/4\},\\ t_{\mathrm{mix}}&=\max_x T(x). \end{aligned}$$

The second quantity is the worst-start mixing time. A typical-start statement instead draws $X_0$ from $\pi$, fixes that realized configuration, and studies $T(X_0)$. Averaging over the initial configuration before measuring distance gives equilibrium at every time, since $\sum_x\pi(x)P_t(x,\cdot)=\pi$. Forgetting a known initial configuration is a different question.

3. Development history and previous partial results

From equilibrium to dynamics

Sherrington and Kirkpatrick introduced their model in 1975 to study magnetism with competing random interactions. Their original paper concerns the equilibrium phases of an infinite-range model.

Parisi’s later description of equilibrium led to a formula for the limiting free energy, a quantity derived from $\log Z$. Guerra’s interpolation bound and Talagrand’s 2006 proof established that formula rigorously. This was a major equilibrium achievement, but a formula for the weights of configurations does not specify how quickly a local dynamics moves among them.

For critical dynamics, numerical work already pointed toward the scale $n^{2/3}$. Billoire and Campbell’s 2011 study, especially its finite-size scaling analysis, examined equilibrium spin autocorrelations. Such evidence suggests a relaxation scale; it does not establish a bound for every initial configuration.

Rigorous progress before this family

High-temperature results gradually extended the range of provably efficient local sampling. Eldan, Koehler, and Zeitouni obtained a spectral criterion giving an SK result for $\beta<1/4$. Later localization methods improved that range. In August 2026, Wang obtained optimal-order $O(n\log n)$ random-site mixing for every fixed $\beta<1/2$, including external fields. In September 2026, Boban, Li, and Oveis Gharan obtained polynomial mixing a fixed distance beyond $1/2$. These advances still left a gap before the zero-field threshold $1$.

The low-temperature story also depends on the starting state. Sellke’s 2025 preprint established exponential worst-case mixing at sufficiently low temperature. It explained why a physics prediction of stretched-exponential relaxation should not be interpreted as a worst-case bound. Rare, exceptionally slow configurations can dominate a maximum without describing an equilibrium sample.

Family 227 claims to close the high-temperature gap, identify the critical mixing exponent, and bound low-temperature relaxation for typical fixed equilibrium states. These are distinct advances with different quantifiers.

4. What the new manuscripts claim

The first six results below concern zero-field Gaussian disorder, with fixed $\beta$ as $n$ grows. Probability tending to one refers to the random couplings and, for typical-start statements, the sampled initial state.

Temperature	Claimed scale in updates per spin	Starting state
High temperature, $0\le\beta<1$	$\log n$, with cutoff	Worst case
Critical, $\beta=1$	$n^{2/3+o(1)}$	Worst case
Low temperature, $\beta>1$	Between two stretched exponentials	Typical fixed Gibbs state

High temperature and cutoff

The spectral-gap manuscript claims a positive relaxation rate independent of $n$ for each fixed $0<\beta<1$. A spectral gap measures how quickly the slowest mode of a reversible dynamics relaxes. This all-observables statement is stronger than controlling only linear combinations of spins.

The cutoff manuscript goes further. With $d_n(t)=\max_x\|P_t(x,\cdot)-\pi\|_{\mathrm{TV}}$, it claims a deterministic rate $\lambda(\beta)>0$ and location

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

For every fixed $0<\varepsilon<1$, it claims $d_n((1-\varepsilon)t_n)\to1$ and $d_n((1+\varepsilon)t_n)\to0$, in probability over disorder. This abrupt drop on the relative time scale is called cutoff. The paper does not identify the smaller cutoff window or give a simple closed formula for $\lambda(\beta)$.

The critical power law

The critical slowing-down manuscript claims that, at any deterministic time $t_n=o(n^{2/3})$, almost all Gibbs-sampled fixed states remain more than $1/4$ from equilibrium. Here $o(n^{2/3})$ means $t_n/n^{2/3}\to0$.

The critical mixing manuscript supplies the claimed matching exponent for the worst start:

$$t_{\mathrm{mix}}=n^{2/3+o(1)}.$$

Precisely, for every fixed $\delta>0$, the mixing time lies between $n^{2/3-\delta}$ and $n^{2/3+\delta}$ with probability tending to one. This does not give an exact constant or critical cutoff. In individual update attempts, the exponent is $5/3$.

Low temperature and typical states

Combining the barrier manuscript with the typical-start upper bound gives, for every fixed $\beta>1$, the claimed bounds

$$\exp\!\left(n^{1/10000}\right) <T(X_0) \le\exp\!\left(n^{1-1/40000000}\right)$$

with joint probability tending to one. Both bounds use rate-one clocks per spin and a fixed realized Gibbs start.

A stretched exponential $\exp(n^a)$, where $0<a<1$, eventually exceeds every polynomial while growing more slowly than $\exp(cn)$ for fixed $c>0$. The enormous gap between the displayed exponents matters: these bounds classify the growth, but do not identify a sharp exponent such as the predicted $1/3$. They also do not bound every starting state.

Universal critical memory curves

The two October manuscripts examine more than mixing time. At equilibrium, define the mean spin autocorrelation

$$C_{n,J}(t)=\frac1n\sum_{i=1}^n \mathbb E_{\pi_{J,1}}[\sigma_i(0)\sigma_i(t)].$$

The expectation averages the equilibrium initial state and the subsequent dynamics, keeping $J$ fixed. The stationary manuscript claims that $n^{1/3}C_{n,J}(s n^{2/3})$, for positive $s$, converges in distribution as a function of time to a random limiting curve. Gaussian couplings and independent fair-sign couplings $J_{ij}=\pm1$ give the same limiting law: this is universality. Individual disorder samples still have different curves.

The quench manuscript starts instead from independent fair spins, then runs the dynamics at $\beta=1$. A quench is this sudden change to the new temperature. The rescaled two-time correlation, with positive waiting time and lag both of order $n^{2/3}$, has a joint universal limit with the stationary curve. In that limit, increasing the waiting time recovers the stationary correlation. This is not a statement of universality for every coupling distribution.

5. Why the advance matters

If correct, the family turns the physical distinction between rapid equilibration, critical slowing, and persistent low-temperature memory into quantitative statements about a canonical random system. It also separates three questions that a single phrase such as “slow mixing” can hide: worst-case sampling, relaxation from typical equilibrium configurations, and the decay of particular observables.

For sampling algorithms, fast mixing gives a way to generate approximately equilibrium-distributed configurations by local updates. Low-temperature typical-start bounds, however, presuppose an equilibrium sample; they do not themselves provide an efficient method for obtaining one.

Importance: 4/5, provisional. The claimed transition across all fixed temperatures, the sharp critical exponent, and the universality results would constitute a major advance in probability and statistical mechanics. The score measures that mathematical significance if the claims hold. It is separate from verification, and will be calibrated against other families as the series grows.

6. The main idea

Independent spins explain the logarithm

At $\beta=0$, every redraw is a fair coin. Starting from all plus spins, the expected total magnetization at time $t$ is $ne^{-t}$: a spin retains its initial bias only if its clock has not rung. Equilibrium fluctuations of the total magnetization have scale $\sqrt n$. Equating these scales gives

$$ne^{-t}\approx\sqrt n \quad\Longrightarrow\quad t\approx\tfrac12\log n.$$

This calculation motivates the cutoff location but is not a full total-variation proof. With interactions, the cutoff manuscript tracks the sensitivity of evolved observables to individual spins. Upper bounds show that all distinguishability decays; lower bounds construct an observable whose initial bias remains detectable. The difficulty is making these estimates uniform over starting configurations.

Collective modes explain critical slowing

Think of diagonalizing a matrix to identify directions that respond weakly to a restoring force. Near criticality, certain collective spin directions become unusually soft. The manuscripts relate those directions to the largest eigenvalues of the normalized interaction matrix, with entries $J_{ij}/\sqrt n$, whose edge fluctuations occur on the scale $n^{-2/3}$.

This suggests a reciprocal time scale $n^{2/3}$, but Ising spins cannot simply be replaced by independent linear modes. The critical proofs compare them with auxiliary spherical spins—real vectors constrained to a sphere—and use progressively revealed noisy observations to control remaining fluctuations. They then transfer functional inequalities back to the discrete dynamics. The stationary and quench papers additionally identify the effective limiting evolution and compare Gaussian with sign disorder. These are proof strategies, not deductions from the eigenvalue scale alone.

Overlap detects barriers

The overlap between configurations is

$$R(\sigma,\tau)=\frac1n\sum_i\sigma_i\tau_i.$$

It equals one for identical configurations and minus one for opposite configurations. A single spin update changes it by at most $2/n$, so local dynamics cannot jump across a macroscopic interval of overlaps.

The lower-bound paper uses constrained equilibrium comparisons and sequences of overlap crossings to obstruct rapid loss of memory for typical states. For the upper bound, the companion paper constructs probabilistic connections between sets of appreciable equilibrium mass, rules out excessively small flow between them, and converts that flow into typical-start mixing. Rare slow states can remain outside this conclusion. The two arguments explain why the lower and upper bounds concern a large fraction of states rather than a maximum over all states.

7. Scope, limitations, and open questions

The pinned release’s family-specific Lean scope document describes partial coverage:

Claim	Coverage stated by the repository
High-temperature spectral gap	Dimension-independent bound for fixed $0<\beta<1$
Cutoff	Discrete-time ratio cutoff for $0\le\beta<1/2$; the full range, explicit location, and continuous-time conclusion are outside this statement
Critical mixing	Worst-start lower bound $n^{2/3-\varepsilon}$ and upper bound $e^{\varepsilon n}$, for fixed $\varepsilon>0$; this does not certify the sharp upper exponent
Critical typical starts and linear observables	The stated pre-$n^{2/3}$ obstruction and covariance/linear-relaxation bounds
Low-temperature barriers	The stated $\exp(n^{1/10000})$ obstruction for typical fixed starts

That document does not list formalizations of the sharp critical upper bound, the low-temperature typical-start upper bound, or the two universality manuscripts. This digest reports the published scope; no Lean build or independent audit of the complete proofs was performed for it.

The temperature is fixed before the system size tends to infinity. These statements do not automatically extend to nonzero fields, temperatures approaching criticality with $n$, all disorder laws, or finite-size numerical predictions. Universality here concerns the specified autocorrelation processes, not every mixing statement in the family. The repository README also explicitly allows for corrections to unformalized results.

8. Sources and further reading

The eight manuscripts

All eight papers are credited to OpenAI. Links below are pinned to the source revision used for this article; the numbered theorem locators are useful entry points.

1. A spectral gap throughout the high-temperature Sherrington–Kirkpatrick phase — September 24, 2026; Theorem 1.1.
2. Cutoff throughout the high-temperature Sherrington–Kirkpatrick phase — September 24, 2026; Theorem 1.1.
3. Critical slowing down in the Sherrington–Kirkpatrick model — September 24, 2026; Theorems 1.2–1.3.
4. Stretched-exponential barriers for typical SK initial states — September 24, 2026; Theorem 1.1.
5. A typical-start upper bound for low-temperature SK Glauber dynamics — September 24, 2026; Theorem 1 and Corollary 2.
6. Critical mixing in the Sherrington–Kirkpatrick model — September 25, 2026; Theorems 1.1–1.2.
7. Functional universality of critical SK autocorrelations — October 5, 2026; Theorem 1.1.
8. Universality of critical quench autocorrelations in the Sherrington–Kirkpatrick model — October 5, 2026; Theorem 1.1.

Historical sources are linked in Section 3. Optional background: Wikipedia’s Markov chain, Gibbs measure, and total-variation distance entries. Essential definitions are given above.

Source revision: adc7f1241b42. Sources consulted: October 8, 2026. Citation data: BibTeX supplied by the eight manuscript directories.

See the series’ AI declaration for authorship and review status.

Back to the family index.