Fluctuations of the Sherrington--Kirkpatrick free energy at critical temperature

Abstract: We consider the Sherrington--Kirkpatrick spin glass model at the critical inverse temperature $β= 1$ with zero external field. We prove that the free energy $F_N = F_{N,β=1}$ of this model has variance [ \mathrm{Var}(F_N) = \frac16 \log N + O(1)\,, ] confirming a physics prediction of Aspelmeier \cite{aspelmeier2008free}, and that the centered and scaled $F_N$ satisfies a Gaussian CLT. We also identify the critical two-replica overlap scale, proving [ \mathbb{E} \langle R_{1,2}^2\rangle \asymp N^{-2/3}\,, ] as conjectured by Talagrand \cite{talagrand2011mean2}, together with a uniform exponential moment bound for $N^{1/3} |R_{1,2}|$. The key input is a comparison between the Ising and spherical SK partition functions $Z_N$ and $Z<sup>{\mathrm{sp}}_N$: if $X_N = Z_N / Z<sup>{\mathrm{sp}}_N$, then $X_N = 1 + o(1)$ in $L<sup>2$. Thus $Z<sup>{\mathrm{sp}}_N$ captures the diverging critical fluctuations of $Z_N$ and serves as a tractable reweighting variable for estimating overlap moments.

• The paper establishes rigorous free energy variance scaling with Var(F_N) = 16 log N + O(1) at β = 1.
• It proves a central limit theorem for the scaled free energy, confirming Gaussian fluctuations at the critical temperature.
• It develops a critical reweighting method by comparing the Ising and spherical SK models, advancing universal fluctuation analysis.

Fluctuations of the Sherrington–Kirkpatrick Free Energy at Critical Temperature

Introduction and Context

The Sherrington–Kirkpatrick (SK) model is a paradigmatic mean-field model for spin glasses, with the random Hamiltonian given by

$$H_N(\sigma) = \sum_{1 \leq i < j \leq N} W_{ij} \sigma_i \sigma_j + \sum_{i=1}^N W_{ii} \sigma_i^2,$$

where $\sigma \in \{\pm 1\}^N$ and $W_{ij} \sim N(0,1/N)$ for $i < j$, $W_{ii} \sim N(0,2/N)$, all independent and symmetric. The central object of interest is the free energy at inverse temperature $\beta$:

$$F_{N,\beta} = \log Z_{N,\beta}, \qquad Z_{N,\beta} = \frac{1}{2^N} \sum_{\sigma} \exp\bigl(\beta H_N(\sigma)\bigr),$$

and the behavior of $F_{N, \beta}$ in the large $N$ limit is a central question in mathematical physics and probability.

The longstanding focus has been on not only the typical thermodynamic limit (i.e., the law of large numbers and Parisi formula for $F_{N,\beta}/N$) but also the subleading fluctuations of $\sigma \in \{\pm 1\}^N$0, which encode fine properties of the Gibbs measure, overlap structure, and the spin glass phase diagram. The critical temperature $\sigma \in \{\pm 1\}^N$1 is a critical point of the SK model's phase transition, and the nature of free energy fluctuations at this point has been a challenging open problem, with predictions from physics (notably Aspelmeier [aspelmeier2008free]) suggesting diverging fluctuations of order $\sigma \in \{\pm 1\}^N$2.

Main Results

This work rigorously establishes two principal results about the SK free energy at critical $\sigma \in \{\pm 1\}^N$3:

• Asymptotic variance: The variance of the free energy diverges logarithmically:

$\sigma \in \{\pm 1\}^N$4

confirming Aspelmeier's physics prediction and settling conjectures by Talagrand [talagrand2011mean2] and subsequent mathematical work.

• Central Limit Theorem: The centered, scaled free energy exhibits a Gaussian limit:

$\sigma \in \{\pm 1\}^N$5

Additionally, the paper determines the precise scaling of the two-replica overlap at criticality:

$\sigma \in \{\pm 1\}^N$6

settling a conjecture of Talagrand and providing evidence for the universality of BBP-type edge statistics in mean-field spin glasses.

Methodology and Technical Innovations

The proof capitalizes on a novel reweighting approach, employing the spherical version of the SK model as an analytic proxy. The model's spherical variant—where spins are constrained to a sphere—admits tractable random matrix representations (in particular, via explicit integral formulas for the partition function in terms of the spectrum of the disorder matrix). While previous second moment methods have relied on comparison to the mean or low-dimensional statistics, these break down at criticality where all quenched moments diverge.

The principal insight is to compare the Ising SK partition function $\sigma \in \{\pm 1\}^N$7 with its spherical analogue $\sigma \in \{\pm 1\}^N$8:

$\sigma \in \{\pm 1\}^N$9

A central technical result establishes that $W_{ij} \sim N(0,1/N)$0 in $W_{ij} \sim N(0,1/N)$1, even though both $W_{ij} \sim N(0,1/N)$2 and $W_{ij} \sim N(0,1/N)$3 have diverging variance. This "critical reweighting" leverages the spherical model's explicit contour integral formula and corresponding random matrix theory to transfer precise fluctuation and overlap control from the spherical to the Ising setting.

BBP-type critical edge scaling results, originating in random matrix theory [baik2005phase, bloemendal2013limits], enter crucially: the localization of overlap fluctuations at the scale $W_{ij} \sim N(0,1/N)$4 arises from the analysis of the joint distribution of edge eigenvalues under spiked perturbations, connecting the structure of the SK spin glass to universality at the edge of Wigner semicircle ensembles.

Direct Numerical and Probabilistic Claims

• Variance scaling: $W_{ij} \sim N(0,1/N)$5.
• Overlap scaling: $W_{ij} \sim N(0,1/N)$6; for the exponential moment $W_{ij} \sim N(0,1/N)$7.
• Spherical model: The same leading fluctuation and overlap estimates apply to the spherical SK model, with analogous variance and exponential overlap moment bounds.

These results are the first to provide the precise asymptotic scaling for both the variance and the overlap moments at the critical temperature in the SK model and its spherical counterpart.

Theoretical and Practical Implications

• Universality at criticality: The results highlight the validity of universality conjectures—namely, that fluctuations in the SK model's free energy at the phase transition are governed by BBP-type edge statistics, as in random matrix theory.
• Critical reweighting methodology: The successful implementation of critical reweighting via the spherical partition function opens the door to analyze other models and regimes where the standard second-moment method fails due to diverging moments.
• Overlap and Gibbs geometry: The explicit identification of the critical overlap scaling provides new information about replica symmetry breaking and the organization of states close to the phase transition, which is essential for both the mathematical theory and numerical studies of glassy systems.

Relation to Previous and Concurrent Works

• Comparison to prior bounds: The new results drastically improve over previous bounds on variance and overlap moments: Chatterjee's $W_{ij} \sim N(0,1/N)$8 upper bound and Chen--Lam's $W_{ij} \sim N(0,1/N)$9 bound for the free energy variance [chen2019order].
• Simultaneous and independent work: Schertzer [schertzer2026order] provides sharp bounds for the variance at criticality; the present work achieves the exact scaling with matching upper and lower bounds.
• Broader context: The methods clarify the differences and relations between the SK model and other mean-field models at criticality, e.g., critical Ising models on random graphs [prodromidis2026distribution], and suggest the methods can be applied or adapted in broader contexts, including bipartite and multi-species models.

Future Directions

Several directions are implicated for further research:

• Fluctuations beyond criticality: The paper suggests extending the critical reweighting techniques to temperatures in the critical window $i < j$0, where the limiting distribution is expected to be a mixture of Gaussian and Tracy-Widom laws, but the random matrix analysis required is more subtle.
• Multi-species and bipartite models: The analytic framework may transfer to "vector" spin glasses or those with more complex order parameter structure, with appropriate adaptations to the reweighting and comparison methods.
• Non-Gaussian disorder and universality: An intriguing question is whether and how much these results depend on the Gaussianity of the disorder—aspects of universality could be tested rigorously under perturbations.

Conclusion

This paper establishes the precise scaling of free energy fluctuations and overlap moments for the critical ($i < j$1) SK model, using a novel and robust comparison to the spherical model via critical reweighting. This resolves long-standing conjectures regarding the order of fluctuations at the phase transition and introduces techniques of broad applicability in the study of mean-field disordered systems. The results also reinforce the deep connections between spin glass theory and random matrix edge behavior, notably BBP transitions, thus advancing both fields and linking their fluctuation theories at the finest scales.

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References:

• T. Aspelmeier, "Free-energy fluctuations and chaos in the Sherrington–Kirkpatrick model" [aspelmeier2008free].
• P.S. Dey and T. Kang, "Fluctuations for the Sherrington–Kirkpatrick spin glass model near the critical temperature" (Dey et al., 5 Mar 2026).
• B. Landon, "Free energy fluctuations of the two-spin spherical SK model at critical temperature" [landon2022free].
• S. Chatterjee, "Disorder chaos and multiple valleys in spin glasses" [chatterjee2009disorder].
• W.-K. Chen and W.-K. Lam, "Order of fluctuations of the free energy in the SK model at critical temperature" [chen2019order].
• M. Talagrand, "Mean Field Models for Spin Glasses" [talagrand2010mean], [talagrand2011mean2].
• A. Schertzer, "The order of free energy fluctuations in the critical Sherrington–Kirkpatrick model revisited" (Schertzer, 19 Jun 2026).
• K.-I. Prodromidis and A. Sly, "Distribution of the magnetization of the critical Ising model on sparse random graphs" (Prodromidis et al., 30 Mar 2026).
• J. Baik, G. Ben Arous, and S. Péché, "Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices" [baik2005phase].
• A. Bloemendal and B. Virág, "Limits of spiked random matrices" [bloemendal2013limits].