The low-temperature Sherrington–Kirkpatrick fluctuation law

1 · The essentials

The Sherrington–Kirkpatrick model describes a random system whose overall behavior changes from one random realization to another. These manuscripts study the size and pattern of those changes as the system grows. They focus on a quantity called the log partition function, under the specific assumptions of Gaussian randomness, no external field, and a fixed inverse temperature greater than one. The main manuscript reports that the typical spread, measured by the standard deviation, grows like the system size raised to the one-sixth power, multiplied by a positive constant. It also reports that subtracting the exact average and dividing by the exact standard deviation produces a distribution that settles toward a definite shape. This convergence applies as the size runs through all integers, and the limiting shape is not concentrated at a single value. These results are shown in the manuscript, but the main result is not formalized in Lean and has not been formally verified.

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

As an imperfect image, imagine comparing batches of randomly filled bags, first subtracting each batch’s average weight and then measuring differences in units of that batch’s spread. The claim here is that, after an analogous adjustment, the pattern of differences approaches a fixed shape as the system grows; the bag image does not capture the model’s interactions.

Why it matters

The source identifies the one-sixth fluctuation exponent as a prediction, and these manuscripts report results supporting it in the stated low-temperature regime. The main result goes beyond an exponent: it gives a finite positive variance coefficient and convergence to a nondegenerate limiting law. This distinguishes knowing how large fluctuations are from knowing their eventual distribution.

2 · Going further

The precise setting is the zero-field Gaussian Ising Sherrington–Kirkpatrick model at every fixed inverse temperature $β > 1$ . Writing $L_{n}$ for the log partition function at system size $n$ , the repository summary states that $Var (L_{n}) / n^{1/3} \to c_{β}$ , where $0 < c_{β} < \infty$ . Consequently, its standard deviation is asymptotic to $c_{β} n^{1/6}$ .

The main manuscript, titled “The low-temperature Sherrington–Kirkpatrick free-energy limiting law,” also reports convergence in distribution after exact centering and standardization: $(L_{n} - E L_{n}) / Var (L_{n})$ converges to a nondegenerate law. The convergence is along all integer system sizes, rather than only selected subsequences. The repository summary describes the limiting law as uniquely characterized, but the supplied abstract does not give that characterization or an explicit formula for it.

The companion manuscript isolates the fluctuation scale. It reports a standard deviation of $n^{1/6 + o (1)}$ and states that the same exponent governs typical centered absolute fluctuations. The main manuscript supplies a sharper variance asymptotic and a limiting distribution. Neither supplied abstract describes the argument’s mechanism, so the source does not support an account of the techniques used to obtain these conclusions.

Useful notions

Expectation and centering
Variance and standard deviation
Convergence in distribution
Asymptotic growth rates
Partition functions in statistical mechanics

What was known before / what is new

Before

The source describes the one-sixth standard-deviation exponent as predicted. It does not identify who made that prediction or give earlier fluctuation bounds or limiting-law results.

New

The companion manuscript reports the predicted exponent for both the standard deviation and typical centered absolute fluctuations at every fixed $β > 1$ . The main manuscript reports the sharper asymptotic $Var (L_{n}) \sim c_{β} n^{1/3}$ with a finite positive coefficient. It also reports full-sequence convergence of the exactly centered and standardized quantity to a nondegenerate law, described in the repository summary as uniquely characterized.

Limits and assumptions

The stated results concern the zero-field Gaussian Ising Sherrington–Kirkpatrick model with fixed inverse temperature $β > 1$ . The supplied text makes no claim for other temperatures, a temperature changing with system size, nonzero external fields, or other randomness assumptions. It supplies neither an explicit characterization of the limiting law nor details of the argument, and it identifies no remaining open problems. The main result is not formalized in Lean and has not been formally verified.

Mini glossary

Centering: Centering means subtracting a random quantity’s expectation from that quantity.
Standard deviation: The standard deviation is the square root of the variance and measures a random quantity’s spread.
Standardization: Standardization means dividing a centered quantity by its standard deviation.
Convergence in distribution: Convergence in distribution means that the probability distributions approach a limiting law.
Nondegenerate law: A nondegenerate law is a probability distribution that is not concentrated at a single value.
Full-sequence convergence: Here, full-sequence convergence means convergence as system size tends to infinity through all integers, not just a selected subsequence.

Keywords : Sherrington–Kirkpatrick model · Fluctuation scale · Limiting law

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 every fixed inverse temperature β > 1, determines the fluctuation scale and limiting law of the zero-field Gaussian Sherrington–Kirkpatrick log partition function. Its variance is asymptotic to $c_{β} n^{1/3}$ , with $c_{β} > 0$ , confirming the predicted n^1/6 standard-deviation scale. Exact centering and standardization give full-sequence convergence to a uniquely characterized nondegenerate law.

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

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The low-temperature Sherrington–Kirkpatrick free-energy limiting law

24/09/2026

To be verified

For every fixed inverse temperature β > 1, we prove that the zero-field Gaussian Ising Sherrington–Kirkpatrick free energy, centered by its expectation and divided by its standard deviation, converges in distribution to a nondegenerate law as the system size tends to infinity through all integers. We also prove that its variance divided by n^1/3 converges to a finite positive constant.

PDF GitHub folder

The low-temperature Sherrington–Kirkpatrick fluctuation scale

24/09/2026

To be verified

For the zero-field Gaussian Sherrington–Kirkpatrick model at every fixed inverse temperature β > 1, we prove that the standard deviation of the log partition function is $n^{1/6 + o (1)}$ . The same exponent describes its typical centered absolute fluctuations, establishing the predicted one-sixth exponent in this regime.

PDF GitHub folder