Low-temperature SK free-energy fluctuations

Source and verification

This article explains family 217 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

A formula for the limiting free energy does not determine fluctuations between disorder samples. Family 217 claims the low-temperature one-sixth fluctuation scale for the Gaussian Sherrington–Kirkpatrick model, followed by an exact variance prefactor and a full-sequence limiting law.

🏷️ Preliminaries: disorder, pressure, and replicas

Let $g_{ij}$, $1\le i<j\le n$, be independent standard Gaussians. For Ising spins ${\sigma }_i\in \{-1,1\}$, set

$$\begin{array}{rl}{H}_n(\sigma )& =n^{-1/2}\sum _{i<j}{g}_{ij}{\sigma }_i{\sigma }_j,\\ Z_n(\beta )& =\sum _{\sigma }{e}^{\beta H_n(\sigma )},\\ F_n& =\log Z_n(\beta ).\end{array}$$

The disorder is the collection of couplings $g_{ij}$. Conditional on these couplings, the Gibbs law samples spins with probabilities proportional to their weights. Two independent samples from this same conditional law are called replicas. Their overlap is $R(\sigma ,\tau )=n^{-1}{\sum }_i{\sigma }_i{\tau }_i$.

The thermodynamic pressure is $n^{-1}\mathbb{E}{F}_n$. Its limit describes the extensive mean contribution. Here the object is instead $F_n-\mathbb{E}{F}_n$, where expectation averages the disorder. Dividing the spin sum by $2^n$ subtracts the deterministic constant $n\log 2$ and leaves these centered fluctuations unchanged.

🏷️ The two levels of conclusion

Fixed low temperature

For every fixed $\beta >1$, the earlier companion claims

$$\begin{array}{rl}\mathrm{Var}(F_n)& =n^{1/3+o(1)},\\ \mathrm{sd}(F_n)& =n^{1/6+o(1)}.\end{array}$$

It also places the typical centered magnitude between $n^{1/6-\eta }$ and $n^{1/6+\eta }$ with probability tending to one, for each $\eta >0$. (OpenAI, 2026)

The later companion claims a constant $c_{\beta }\in (0,\infty )$ and a nondegenerate law ${\nu }_{\beta }$ such that

$$\begin{array}{rl}{n}^{-1/3}\mathrm{Var}(F_n)& ⟶c_{\beta },\\ \frac{F_n-\mathbb{E}{F}_n}{\sqrt{\mathrm{Var}(F_n)}}& ⟹{\nu }_{\beta }.\end{array}$$

The convergence is through all integers, and the law is characterized by finite-system moment experiments. (OpenAI, 2026b)

The papers do not identify this law as Gaussian or as a Tracy–Widom law. Their statements keep $\beta$ fixed; they do not automatically cover a temperature approaching the critical value or zero temperature.

🏷️ A normalization check and a baseline estimate

For the physical free-energy density $-F_n/(\beta n)$, the standard-deviation exponent is $1/6-1=-5/6$. Confusing total free energy with density would therefore reverse the apparent growth or decay.

Gaussian Poincaré gives a useful elementary benchmark. Since

$$\frac{\partial F_n}{\partial g_{ij}}=\frac{\beta }{\sqrt{n}}⟨{\sigma }_i{\sigma }_j⟩,$$

and $|⟨{\sigma }_i{\sigma }_j⟩|\le 1$,

$$\begin{array}{rl}\mathrm{Var}(F_n)& \le \sum _{i<j}\mathbb{E}{\left|\frac{\partial F_n}{\partial g_{ij}}\right|}^2\\ & \le \frac{{\beta }^2(n-1)}{2}.\end{array}$$

This only gives standard deviation of order at most $n^{1/2}$. The claimed $n^{1/6}$ scale requires much more information about correlated Gibbs responses.

🏷️ Why the limiting law needs a new comparison

The scale paper organizes Gaussian covariance into levels. At a level with averaging parameter $u$, the recursion uses $u^{-1}\log \mathbb{E}{e}^{uX}$, interpreted as ordinary expectation at $u=0$. Replicas share initial Gaussian history and then branch. Overlap estimates compare this finite hierarchy with scalar information from the Parisi minimizer.

To obtain an exponent, suitably controlled Laplace windows and nonconcentration estimates suffice. To obtain a limiting law, the later paper compares sizes $N$ and every $N^{\prime }\in [N,2N]$ after the normalization $n^{-1/6}$. Its central improvement is an error summable over dyadic size intervals. Moment comparisons then make all subsequences agree.

Four parts of the source outline are especially consequential. Covariance estimates must remain uniform when hierarchy levels become small. Higher moments must be proved under the actual conditioned comparison laws. Signed pressure terms must be combined before taking absolute values so that leading terms cancel. Finally, positive Laplace tests need tail control before they can yield convergence of ordinary moments and a moment-determinate law.

The resulting moment characterization is an identification of the law, even without a familiar density formula. It is stronger than tightness, stronger than a variance equivalent, and stronger than convergence along one selected subsequence.

🏷️ Key steps and proof outlines

The exponent calculation and the full limiting-law calculation share covariance interpolation, but their final estimates differ. A subpower uncertainty is sufficient for an exponent; identifying one law requires summable comparisons between sizes.

Gaussian completion removes a covariance constant

With $T={\beta }^2$ and the factor $\beta$ included in the Hamiltonian, the covariance of two spin configurations is

$$\mathbb{E}{H}_n(\sigma )H_n(\tau )=\frac{nT}{2}R(\sigma ,\tau {)}^2-\frac{T}{2}.$$

Add the same independent Gaussian $D$ of variance $T/2$ to every spin energy. Its only effect on the log partition function is $F_n^c=F_n+D$, and the covariance becomes exactly $nT{R}^2/2$. This is useful because the hierarchical interpolation is expressed in terms of this overlap square. At a Laplace argument $t$, the correction is explicit:

$$\log \mathbb{E}{e}^{t(F_n^c-\mathbb{E}{F}_n^c)}=\log \mathbb{E}{e}^{t(F_n-\mathbb{E}{F}_n)}+T{t}^2/4.$$

Thus the completion can be removed at the end; it is not an uncontrolled change of model.

Completing a square turns interpolation into a comparison

The pressure functional $f_n(b,h)$ uses a matrix-covariance path $b$ and a field-covariance path $h$, indexed by hierarchy mass $u\in [0,1]$. Its differentiated formula contains overlap quantities $S(u)$ and $B(u)$ satisfying $B(u)\ge S(u{)}^2$. Here $S$ is the expected overlap of replicas sharing history to mass $u$, and $B$ its second moment. For admissible endpoints $(b,h)$ and $(b+a,h-c)$, with $a\ge 0$ and $c=0$ wherever $a=0$, the source obtains

$$\begin{array}{rl}\frac{d}{dt}{f}_n(b+ta,h-tc)& =\frac{1}{4}\int (-aB+2cS)du\\ & \le \frac{1}{4}\int \frac{c^2}{a}du.\end{array}$$

The inequality uses

$$-a{S}^2+2cS=\frac{c^2}{a}-a(S-c/a{)}^2$$

where $a>0$, and uses zero for the quotient where $a=c=0$. Integrating gives the chord comparison. Admissibility is a condition on both endpoints and their segment; a difference of admissible paths is not automatically an admissible covariance path.

In the scale paper, move the matrix disorder from hierarchy mass zero to a small positive mass $s=n^{-1/6+5m}$, for fixed small $m>0$. If $p(\epsilon )$ is the corresponding per-site pressure along this move, the endpoint identity is

$$K_n(s):=\log \mathbb{E}{e}^{s(F_n^c-\mathbb{E}{F}_n^c)}=ns[p(T)-p(0)].$$

The optimized-profile estimate and chord comparison give, for $\epsilon =n^{-1/3}$, a gain of order $\epsilon s^3=n^{-5/6+15m}$ and an error of order $n^{-5/6+7m+20\delta }$ with $\delta =m/10000$. The strict exponent margin makes the gain survive. Multiplication by $ns$ gives $K_n(s)\gtrsim n^{20m}$. The upper comparison adds a low-field barrier, bounds the low-overlap mean, and integrates susceptibility estimates over dyadic mass intervals; its power ledger gives $K_n(s)\lesssim n^{100m}$. These Laplace windows, together with the source’s tail and nonconcentration arguments, yield the stated exponent results (OpenAI, 2026a).

A stopped change of measure controls actual tilted laws

For the later comparison, put $x=N^{-1/6}$ and use small positive Laplace arguments $s=q{x}^{1+\epsilon }$. The interpolation estimates are proved while the centered log Laplace transform is at most $3x^{\epsilon }$. This is a restriction on deterministic path data after disorder expectation, not a random stopping event in a disorder sample.

A key device adds independent exponential noise of rate $x/2$ to the intermediate continuation $G_m$. After pinning one spin, the Gaussian-driver density has negative coefficients on convex continuation functions; the exponential augmentation leaves these coefficients nonnegative in magnitude and restricts to a convex epigraph. Marginalization produces a log-concave one-dimensional density. Its two-sided tail bound, combined with the small positive Laplace transform, bounds the variance scale and then every fixed $L^p$ norm of

$$\frac{e^{s{G}_m}}{\mathbb{E}{e}^{s{G}_m}}.$$

Hölder’s inequality can therefore transfer sufficiently high moments from the untilted law to the actual tilted comparison law. This is the justification for changing measures inside later covariance estimates.

A discrete derivative is needed when the system size changes

The number of spins is integer-valued, so differentiating only the continuous covariance parameters misses a term. The source smooths over neighboring integer sizes with a window of width $D=x^{-1/2}$ and uses coefficients $c_j$ satisfying

$$\sum _{j=0}^r{c}_{j}P(j)=P^{\prime }(0)\text{for every polynomial of degree at most }r.$$

Moving this stencil onto the smooth averaging weights recovers their derivative, with remainder $O(D^{-r-1})$ times the supremum of the tested sequence. The covariance and pressure expansions must bound the signed sum of the clock derivative and this size derivative before absolute values are taken. Their leading terms cancel there.

After choosing Taylor orders, loss exponents, and cutoffs in the source’s prescribed order, the remaining increment saves a fixed power. It yields, uniformly for every $N^{\prime }\in [N,2N]$, a comparison of log Laplace transforms of $Y_n=n^{-1/6}(F_n^c-\mathbb{E}{F}_n^c)$ of size $O(x^{\epsilon +c^{\prime }})$ for some $c^{\prime }>0$ (OpenAI, 2026b).

Summable moment differences identify one nondegenerate law

Fix a desired moment order $k$, then choose $\epsilon$ with $k\epsilon <c^{\prime }/2$. Interpolation at $0,x^{\epsilon },\dots ,k{x}^{\epsilon }$ recovers cumulants through order $k$ with loss at most $x^{-k\epsilon }$. Analytic Taylor remainders are controlled by the preliminary exponential-tail scale. The resulting dyadic cumulant differences still save a positive power, so their sum over dyadic sizes converges. Uniformity for every size inside a dyad gives full-sequence convergence.

The limiting variance cannot vanish: its power-rate convergence to zero would contradict the earlier lower bound $\mathrm{Var}(Y_n)\ge n^{-\eta }$ for every fixed $\eta >0$ eventually. Bounded variance improves the augmented log-concave tails to a common exponential scale. Conditional Jensen removes the independent centered noise, giving uniform exponential moments of $Y_n$. These imply tightness, uniform integrability of all powers, and moment determinacy. Every subsequential law therefore agrees. Finally, the scaled Gaussian completion tends to zero in $L^2$, and normalization by the exact finite-size standard deviation gives the stated law ${\nu }_{\beta }$.

See Also

• Random matrices and spin-glass complexity studies counts of critical points in spherical models, a different observable from the Ising SK partition function.
• Family 221 introduces hierarchical cavity messages for sparse interactions.
• Family 222 uses overlap hierarchies for nonlinear random constraints.

🏷️ Manuscripts and verification scope

• The low-temperature Sherrington–Kirkpatrick free-energy limiting law — pinned manuscript and companion source.
• The low-temperature Sherrington–Kirkpatrick fluctuation scale — pinned manuscript and companion source.

No family scope page was present at lean/docs/217.md in the pinned snapshot. This does not establish that all related formalization code is absent. 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. The low-temperature Sherrington–Kirkpatrick fluctuation scale.

🐻  OpenAI 2026b. The low-temperature Sherrington–Kirkpatrick free-energy limiting law.