Critical SK autocorrelation processes and dynamics across the temperature transition

1 · The essentials

These manuscripts study how a system of randomly interacting spins loses memory of its starting arrangement. The model has no applied field, and each spin has its own clock for updates. On the warmer side of the temperature transition, the approach to equilibrium happens abruptly on a time scale growing like the logarithm of the number of spins. At the transition, the time needed instead grows roughly like the number of spins raised to the two-thirds power. On the colder side, the manuscripts give much longer lower and upper time bounds for a typical equilibrium arrangement chosen once and then held fixed as the starting state. At the transition, two different choices of random interactions give the same limiting description of how spins remember their past, although that description still varies randomly between samples. The family includes a Lean-formalized result establishing a critical slow-mixing lower bound; the supplied formalization scope does not cover all these conclusions.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine a room of people repeatedly reconsidering a two-way choice while responding to randomly assigned influences from others. The question is how long the room remembers its original choices, and whether beginning with independent choices eventually produces the same memory pattern as beginning in equilibrium. This image illustrates the comparison of starting conditions, not the mathematical update rule.

Why it matters

The family connects the time needed to approach equilibrium with the persistence of spin memory across a temperature transition. Its critical results distinguish a common limiting law from a deterministic outcome: Gaussian and Rademacher interactions share the same law, but sample-to-sample randomness survives. The separate treatment of worst-case starts and typical equilibrium configurations also makes clear that the starting-state question matters.

2 · Going further

The setting is zero-field Sherrington–Kirkpatrick heat-bath dynamics with rate-one clocks at each of $n$ sites. The central autocorrelation problem concerns the critical inverse temperature $β = 1$ : multiply mean spin autocorrelations by $n^{1/3}$ and measure waiting times and lags in units of $n^{2/3}$ . Manuscript 1 shows in the manuscript joint functional convergence of stationary and quench autocorrelations for Gaussian and Rademacher couplings to the same random limit. The quench begins with independent fair spins, and convergence is uniform on compact sets of positive waiting times and lags. As the waiting time tends to infinity, its limiting autocorrelation relaxes to the stationary limit.

Manuscript 2 supplies the stationary counterpart: convergence in law is uniform on compact positive-time intervals, the limiting law is not a point mass, and its functions decay to zero at large times. The abstracts describe the central comparison—matching the rescalings for two interaction distributions and two preparations—but do not specify the argument's technical mechanism. Accordingly, they support an account of the limiting statements, not a reconstruction of their proofs.

The temperature-dependent companion manuscripts concern mixing and relaxation. For fixed $0 < β < 1$ , Manuscript 3 shows in the manuscript that the unscaled spectral gap stays bounded away from zero with probability tending to one over Gaussian disorder, equivalently giving a dimension-free Poincaré inequality. For fixed $0 \leq β < 1$ , Manuscript 4 shows in the manuscript worst-start total-variation cutoff at $lo g n / (2 λ (β))$ , with $λ (β) > 0$ deterministic. Manuscript 8 gives the critical worst-start mixing scale $n^{2/3 + o (1)}$ , or $n^{5/3 + o (1)}$ uniform-site update attempts, in probability over disorder. The Lean-formalized Manuscript 5 establishes the complementary critical lower bound: for every deterministic $t_{n} = o (n^{2/3})$ , the Gibbs mass of fixed initial states whose distance from equilibrium at $t_{n}$ exceeds $1/4$ tends to one in probability over disorder; the discrete counterpart uses $k_{n} = o (n^{5/3})$ .

For fixed $β > 1$ , Manuscript 6 shows in the manuscript that the Gibbs mass of fixed starting states still farther than $1/4$ from equilibrium at time $exp (n^{1/10000})$ tends to one in probability over disorder. It states this both for rate-one-per-site time and for the same number of discrete update attempts. Manuscript 7 supplies a typical-start upper bound in the zero-field Gaussian model: distance is at most $1/4$ by time $exp (n^{1 - 1/40000000})$ , with joint probability tending to one over disorder and the Gibbs-sampled starting state. Sampling an equilibrium configuration and then conditioning on that fixed configuration is essential to these statements.

Useful notions

Random spin interactions and disorder
Equilibrium and the Gibbs distribution
Continuous-time single-site updates
Autocorrelation and rescaling
Convergence in law and total-variation distance

What was known before / what is new

Before

The supplied source gives no historical account of earlier results or open problems. Within this family, the stationary critical autocorrelation manuscript provides the counterpart to the joint stationary-and-quench treatment. The high-temperature spectral-gap and cutoff manuscripts provide the warmer-phase comparison.

New

Manuscript 1 shows in the manuscript that critical stationary and quench autocorrelations have a common functional scaling limit across Gaussian and Rademacher couplings. It also identifies the quench limit selected by independent fair starting spins and its relaxation to the stationary limit. The companion manuscripts describe logarithmic high-temperature cutoff, the critical worst-start mixing exponent, and separated low-temperature typical-start bounds.

Limits and assumptions

The results concern zero-field SK single-site heat-bath dynamics, with time conventions that must be kept distinct. The critical autocorrelation universality statements cover Gaussian and Rademacher couplings and compact sets of strictly positive scaled times; they do not assert a deterministic limit. The high-temperature claims hold for fixed inverse temperature below one, and the low-temperature bounds hold for fixed inverse temperature above one and typical Gibbs-sampled configurations held fixed, not arbitrary worst-case starts. The supplied Lean-formalized scope is Manuscript 5's critical typical-fixed-start lower bound, not the joint autocorrelation universality, the critical upper bound, or the full temperature-dependent collection. The low-temperature lower and upper bounds do not match, and the source supplies neither an exact low-temperature mixing scale nor an explicit list of remaining open problems.

Mini glossary

Disorder: The random interaction couplings specifying a sample of the spin system.
Autocorrelation: A measure of how a spin's value at one time is related to its value at a later time.
Quench: Here, dynamics started from independent fair spins rather than from equilibrium.
Gibbs distribution: The equilibrium distribution from which the typical fixed starting configurations in these results are sampled.
Total-variation distance: The measure of distance from equilibrium used to state the mixing bounds.
Cutoff: An abrupt transition toward equilibrium near a specified mixing-time location.

Keywords : SK dynamics · Critical autocorrelation · Mixing times

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

For zero-field Gaussian SK heat-bath dynamics with rate-one updates per spin, proves worst-start cutoff on the $lo g n$ scale for fixed $0 \leq β < 1$ , 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.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/227.md

Manuscripts in this family

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Universality of critical quench autocorrelations in the Sherrington–Kirkpatrick model

05/10/2026

To be verified

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 GitHub folder

Functional universality of critical SK autocorrelations

05/10/2026

To be verified

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 GitHub folder

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

24/09/2026

To be verified

For every fixed inverse temperature $0 < β < 1$ , 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 GitHub folder

Cutoff throughout the high-temperature Sherrington–Kirkpatrick phase

24/09/2026

To be verified

We prove worst-case total-variation cutoff for the zero-field Gaussian Sherrington–Kirkpatrick heat-bath dynamics at every fixed inverse temperature $0 \leq β < 1$ . With rate-one refresh at each spin, the cutoff location is $lo g n / (2 λ (β))$ for a positive deterministic rate $λ (β)$ . The location for uniformly chosen single-site update attempts is n times as large. Convergence is in probability over the disorder.

PDF GitHub folder

Critical slowing down in the Sherrington–Kirkpatrick model

24/09/2026

Verified by 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 GitHub folder Lean formalization

Stretched-exponential barriers for typical SK initial states

24/09/2026

To be verified

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 GitHub folder

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

24/09/2026

To be verified

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 GitHub folder

Critical mixing in the Sherrington–Kirkpatrick model

25/09/2026

To be verified

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 GitHub folder