The low-temperature Sherrington–Kirkpatrick fluctuation law

The typical spread of the low-temperature Sherrington–Kirkpatrick free energy grows like n^(1/6), with a limiting law.

2 papers September 24, 2026 Explained by AI

The Sherrington–Kirkpatrick model is a standard model of a disordered magnet with random couplings. Its free energy changes from one random sample to the next. Physicists predicted that, at low temperature, the typical change grows like n to the power 1/6. These two papers claim to prove that scale, then a positive variance constant and a limiting shape after rescaling. No computer proof checker has checked them.

First step

1992

Crisanti, Paladin, Sommers and Vulpiani, the n^(-5/6) prediction

Significance

Major

If it holds, it settles a long-standing physics prediction for the fluctuation scale and adds a limiting law.

Trust

Claimed, not checked

Field

Probability and statistical mechanics

2 papers

The explanation

Take two magnets, each pointing up or down. Write their states as +1 or −1. A random number g, drawn from a bell curve, decides whether they prefer to agree or disagree. At inverse temperature 2, the partition function adds a weight for each of the four states. Here it comes to 4 cosh(√2 g). Its logarithm is the free energy. For g = 0 it is log 4, about 1.386. For g = 1 or g = −1 it is about 2.165. For g = 2 it is about 3.525. So the free energy is itself random, because it depends on the drawn couplings.

The Sherrington–Kirkpatrick model does this with n magnets, often called spins. Every pair of spins gets its own independent random coupling, drawn from a bell curveSource 1. The couplings are scaled by one over the square root of n. Inverse temperature, written β, sets how strongly the system prefers low-energy states. Above β = 1 the model is in its low-temperature, glassy phase. The question here is about the sample-to-sample fluctuations. Draw the couplings again, and the free energy moves. By how much, and with what shapeSource 2?

The average free energy is well understood. Parisi proposed a formula for its leading term in 1979. Guerra and Talagrand made that formula rigorous by 2006Source 3. Those results leave the smaller fluctuations around the average. Physicists predicted a free energy per spin varying on the scale n to the −5/6. That becomes n to the 1/6 for the whole log partition functionSource 4. The prediction rests on a matching argument that is heuristic. It gives neither a variance formula nor a limiting lawSource 5.

Rigorous bounds were far from that. In 2026, Aronow and Lopatto bounded the variance between constant multiples of n to the 4/15 and n to the 7/15Source 6. Picture a million spins. Then n to the 4/15 is about 39.8, and n to the 7/15 is about 631. The predicted variance, n to the 1/3, is 100. So the predicted spread, its square root, is 10. The bounds allowed this value but did not establish it.

The fluctuation-scale paper claims the predicted exponent for every fixed β above 1. It allows an arbitrarily small error in the powerSource 7. The limiting-law paper goes further. It claims that the variance divided by n to the 1/3 tends to a positive constantSource 8. After exact centering and scaling, the free energy then tends to one fixed bell-like law, called ν_βSource 9. The law has mean zero and variance one. It is fixed by a list of moments, which are averages of powers. The paper does not identify it with any named distributionSource 10.

The obstacle is comparing systems of different sizes. An exponent with slack does not show that the variance settles to a constant. That needs errors which add up to a finite total over all sizesSource 11. The method reveals the random couplings in stages, like a branching tree of shared histories. It then compares neighbouring sizes through that tree. A key input is a 2026 result of Lopatto on the shape of Parisi's solution, used at every β above 1Source 12.

The claims have limits. Each result is for one fixed β above 1, with no external field. Nothing is uniform as β approaches 1 or grows without boundSource 13. The papers also give a related consequence for a model with an added ferromagnetic pull below unit strengthSource 14.

How sure is this? No Lean formalization is listed for these papers in the openai/math release. So no computer proof checker has checked them. No specialist review has been published.

What it doesn't show. This does not name the limiting law or give its variance constant as a number.

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 both papers hold. They settle a predicted scale and add a law, but no Lean formalization is listed and no specialist review is published.

Words used

Spin: A tiny magnet that points either up (+1) or down (−1).
Partition function: The sum of the weights of all states of a system; its logarithm is the free energy used here.
Inverse temperature: A number β that sets how strongly the system favours low-energy states; larger means colder.
Free energy fluctuations: How much the free energy changes when the random couplings are drawn again.
Variance: The average squared distance from the mean; its square root is the standard deviation, or typical spread.
Limiting law: The fixed probability shape that a rescaled random quantity approaches as the system grows.
Moments: Averages of the first, second, third and higher powers of a random quantity.

The original papers (2)

The low-temperature Sherrington–Kirkpatrick free-energy limiting law (PDF, opens in a new tab)
September 24, 2026
The low-temperature Sherrington–Kirkpatrick fluctuation scale (PDF, opens in a new tab)
September 24, 2026

All quoted lines (14)

Let $(g_{ij})_{1\le i<j\le n}$ be independent standard Gaussian variables.
The low-temperature Sherrington–Kirkpatrick free-energy limiting law Back to text
At a fixed temperature below the transition, how large are the sample-to-sample fluctuations of the Sherrington–Kirkpatrick free energy, and do they approach a law after centering and rescaling?
The low-temperature Sherrington–Kirkpatrick free-energy limiting law Back to text
Guerra and Toninelli established the thermodynamic limit by Gaussian interpolation, and Guerra's variational bound and Talagrand's matching result identified its Parisi value.
The low-temperature Sherrington–Kirkpatrick fluctuation scale Back to text
At fixed $\beta$ , this corresponds to $n^{1/6}$ for the log partition function.
The low-temperature Sherrington–Kirkpatrick fluctuation scale Back to text
This matching argument is heuristic: it does not itself provide a variance asymptotic or a limiting law.
The low-temperature Sherrington–Kirkpatrick free-energy limiting law Back to text
For each fixed $\beta>1$ , Aronow and Lopatto prove $c_\beta n^{4/15}\le \Var F_n(\beta)\le C_\beta n^{7/15}$ in, and quantitative tail estimates in a separate deviation window in.
The low-temperature Sherrington–Kirkpatrick free-energy limiting law Back to text
Theorem … establishes the predicted one-sixth exponent with arbitrarily small power slack, for every fixed finite $\beta>1$ .
The low-temperature Sherrington–Kirkpatrick fluctuation scale Back to text
The quantitative statement below proves that the variance has a positive limiting prefactor and provides the rate needed for the moment characterization.
The low-temperature Sherrington–Kirkpatrick free-energy limiting law Back to text
We prove a limiting law for every fixed inverse temperature $\beta>1$ in the zero-field Gaussian Ising model.
The low-temperature Sherrington–Kirkpatrick free-energy limiting law Back to text
The formula below specifies the moments of $\nu_\beta$ ; no identification with a named distribution is made.
The low-temperature Sherrington–Kirkpatrick free-energy limiting law Back to text
The additional work here is a quantitative comparison of different sizes, with a summable error.
The low-temperature Sherrington–Kirkpatrick free-energy limiting law Back to text
We use Lopatto's full-support result for every fixed $\beta>1$ , in the form developed in PE.
The low-temperature Sherrington–Kirkpatrick free-energy limiting law Back to text
The temperature is fixed throughout, with no uniform conclusion as $\beta$ approaches $1$ or infinity.
The low-temperature Sherrington–Kirkpatrick fluctuation scale Back to text
This consequence is confined to the displayed zero-field ordered Gaussian model at fixed $\beta>1$ and $0\le\gamma<1$ ; it does not cover $\gamma=1$ or a coupling varying with $n$ .
The low-temperature Sherrington–Kirkpatrick free-energy limiting law Back to text

Other sources

openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model
D. Sherrington and S. Kirkpatrick, “Solvable model of a spin-glass”, Physical Review Letters 35 (1975), 1792–1796
M. Talagrand, “The Parisi formula”, Annals of Mathematics 163 (2006), 221–263
A. Crisanti, G. Paladin, H.-J. Sommers and A. Vulpiani, “Replica trick and fluctuations in disordered systems”, Journal de Physique I 2 (1992), 1325–1332

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