Large-deviation function and small-deviation scaling of the Sherrington–Kirkpatrick free energy: rigorous status of the replica prediction and the matching argument

For the Sherrington–Kirkpatrick (SK) mean-field spin glass, the replica method predicts a complete picture of the sample-to-sample distribution of the free energy: below the critical temperature the replicated potential $\Phi(n,T)=-(\beta n N)^{-1}\ln\overline{Z_J(\beta)^n}$ starts as $\Phi(n,T)=f_{\mathrm{typ}}+c_5 n^5+O(n^6)$ (Kondor's $n^5$ scaling, with $c_5=-(9/640)\beta^5[T\dot q(0)]^3$), which by Legendre duality gives a non-analytic large-deviation rate function $L(f)\propto(f_{\mathrm{typ}}-f)^{6/5}$, and by a non-rigorous matching argument predicts that typical small deviations of the free energy per spin scale as $N^{-5/6}$. Parisi and Rizzo (Phys. Rev. B 79, 134205 (2009)) computed the full $\Phi(n,T)$ phase diagram, proved the $n^5$ scaling at all orders within the hierarchical ansatz, and obtained resummed zero-temperature predictions in agreement with ground-state numerics, while flagging the matching argument, the limit interchange $n\to0$/$N\to\infty$, and the lack of a rigorous derivation as open. Later literature: the critical-temperature variance question was rigorously settled in 2026 preprints ($\mathrm{Var}(F_N(\beta_c))=(1/6)\log N+O(1)$ with a Gaussian CLT, two independent proofs), and a rigorous one-sided large-deviation principle with explicit rate function was established at zero temperature in a 2026 preprint. The predicted near-minimum power law of the rate function was then settled by two further 2026 preprints on the same (upper-tail / positive-$n$) side: Aronow and Lopatto (arXiv:2609.09103) proved two-sided upper-tail estimates with exponent $6/5$ at both positive and zero temperature and bounded the variance of the logarithmic partition function between orders $N^{4/15}$ and $N^{7/15}$, and Pei (arXiv:2610.06480) derived the $q(x)$-dependent amplitude $c_\beta=(9/640)\beta^3\rho_\beta(0)^{-3}$ ($\rho_\beta(0)$ the density at zero of the Parisi measure). What remains open is the exact order of the low-temperature small deviations — whether the matching prediction $N^{-5/6}$ is correct, now bracketed rather than identified — and any finite-temperature rigorous theory. The problem asks for a rigorous and quantitatively validated theory of the SK free-energy distribution connecting its large- and small-deviation regimes.

Background

The Sherrington–Kirkpatrick (SK) model is the paradigmatic mean-field spin glass: $N$ Ising spins $\sigma_i\in\{\pm1\}$ interact through a random Hamiltonian $H_J(\sigma)=\sum_{ij}J_{ij}\sigma_i\sigma_j/\sqrt{N}$ with independent standard Gaussian couplings $J_{ij}$. For each disorder realization (sample) $J$, the partition function is $Z_J(\beta)=\sum_\sigma e^{-\beta H_J(\sigma)}$ and the free energy per spin is $f_J=-(\beta N)^{-1}\ln Z_J$. Although $f_J$ self-averages to a typical value $f_{\mathrm{typ}}$ as $N\to\infty$ (given rigorously by the Parisi formula, the celebrated variational expression for this limit), the full probability distribution $P_N(f)$ of $f_J$ across samples encodes the sample-to-sample fluctuations and is far less understood.

Two regimes of $P_N(f)$ are distinguished. Large deviations are the exponentially small tails $P_N(f)\approx\exp[-NL(f)]$ for $O(1)$ differences $f-f_{\mathrm{typ}}$; small deviations are the central peak, whose width shrinks with $N$. The natural tool is the disorder-averaged replicated partition function, summarized by $\Phi(n,T)=-(\beta n N)^{-1}\ln\overline{Z_J(\beta)^n}$: for positive $n$ it is Legendre-dual to the lower tail $L(f)$, and at finite $N$ the function $n\Phi(n)$ generates the cumulants of the extensive free energy $F=Nf$. Kondor (J. Phys. A 16, L127 (1983)) computed $\Phi$ near the critical temperature $T_c$ with the broken replica-symmetry ansatz and found, below $T_c$, $\Phi(n,\beta)=f_{\mathrm{typ}}+c_5 n^5+O(n^6)$: the powers $n^1$ through $n^4$ cancel, a consequence of replica symmetry breaking (RSB), the Parisi mechanism in which the Gibbs state splits hierarchically and is described by a monotone order-parameter function $q(x)$, $x\in[0,1]$; $\dot q(0)$ denotes its derivative at $x=0$. The $n^5$ scaling implies through Legendre duality a non-analytic rate function $L(f)\propto(f_{\mathrm{typ}}-f)^{6/5}$ near $f_{\mathrm{typ}}$, hence non-Gaussian fluctuations; a matching argument — the non-rigorous assumption that the central peak joins smoothly onto this tail — then predicts that small deviations of the free energy per spin scale as $N^{-5/6}$ (equivalently $\mathrm{Var}(F)\sim N^{1/3}$), a prediction that competed with alternative claims of an $N^{-3/4}$ scaling.

The paper Parisi and Rizzo (Phys. Rev. B 79, 134205 (2009)) mapped the phase diagram of $\Phi(n,T)$ in the $(n,T)$ plane, including the de Almeida–Thouless (AT) line — the instability line of the replica-symmetric solution below which RSB is required — proved within the hierarchical (full replica-symmetry-breaking) ansatz that the $n^5$ scaling holds at all orders in perturbation theory, and derived the exact amplitude relation $c_5=-(9/640)\beta^5[T\dot q(0)]^3$. Resumming high-order asymptotic series (Padé approximants, i.e. rational-function approximants of a truncated series) down to $T=0$ gave the quantitative prediction $L(\Delta e)=-1.63250|\Delta e|^{6/5}+3.1|\Delta e|^{8/5}+\cdots$ for the ground-state energy deviations, in remarkable agreement with existing $T=0$ numerical data. The authors explicitly flagged what remains open: the matching argument is not rigorous and appears to fail at the critical temperature (where, as Aspelmeier predicted and later work confirmed, the free-energy variance diverges logarithmically and the limits $n\to0$ and $N\to\infty$ cannot be exchanged); the rigorous footing is partial (the paper cites rigorous results of Talagrand supporting the approach only for positive $n$); and all numerical tests were at $T=0$.

On the rigorous side, the limiting free energy is settled (Parisi formula), and above $T_c$ a Gaussian central limit theorem holds. Below $T_c$ the fluctuations are open: Chatterjee's superconcentration results — superconcentration denoting an upper bound on the variance of a function of many independent Gaussian inputs that improves on the naive Lipschitz bound by a logarithmic factor — give $\mathrm{Var}(F_N)\le CN\log(2+C\beta)/\log N$ for the total free energy at all finite $\beta$ (per spin, order $1/(N\log N)$ up to $\beta$-dependent constants), together with a lower bound of order $N^{-1}$ per spin, so the predicted $N^{-5/6}$ is merely bracketed. A 2025 survey (Collins-Woodfin and Le) states that low-temperature fluctuations remain open and that numerics cannot distinguish $5/6$ from $3/4$. In 2026 preprints the critical-temperature question was resolved: Du and Huang, and independently Y. Chen, proved $\mathrm{Var}(F_N(\beta_c))=(1/6)\log N+O(1)$ with a Gaussian CLT, confirming Aspelmeier's prediction; and Chen, Guionnet, Ko, Lacroix-A-Chez-Toine and Mourrat established, at zero temperature, a rigorous one-sided large-deviation principle for deviations of the maximal energy above its typical value — the same deviation side as the source paper's free-energy lower tail, since at $T=0$ the free energy per spin coincides with the ground-state energy per spin $\min_\sigma H_J(\sigma)/N$, which, the Gaussian coupling distribution being symmetric under $J\to-J$, has the same law as minus the maximal energy per spin — with an explicit variational rate function $\Lambda^*(r)$, proving it is quadratic near its minimum if and only if an external field is present — consistent with the replica picture — while explicitly leaving the predicted near-minimum power law unproven. The spherical SK model — the variant in which the Ising spins are replaced by continuous variables constrained to a sphere, where random-matrix theory makes fluctuations tractable (governed at low temperature by the Tracy–Widom law, the limiting distribution of the top eigenvalue of a large random matrix) — is the solved neighbor that highlights what is missing for Ising SK with its RSB structure.

Problem Statement

Determine whether the replica large-deviation prediction for the Sherrington–Kirkpatrick free energy holds rigorously and can be quantitatively validated as a theory of the full distribution $P_N(f)$ connecting its large- and small-deviation regimes. Concretely, for the SK model at zero external field (Hamiltonian normalized so that the critical inverse temperature is $\beta_c=1$):

1. Near-minimum asymptotics of the rate function. The one-sided large-deviation rate function of the ground-state energy behaves near its minimum at zero field as $\Lambda^*(r)\sim C(r-g_{\mathrm{s}})^{6/5}$ as $r\to g_{\mathrm{s}}^+$, and at finite temperature the rate function of the free energy per spin behaves near its minimum as $L(f)\sim C(f_{\mathrm{typ}}-f)^{6/5}$ as $f\to f_{\mathrm{typ}}^-$ — where $r$ is the maximal energy per spin (the observable of the rigorous zero-temperature large-deviation principle; by the symmetry of the Gaussian coupling distribution under $J\to-J$, minus the maximal energy per spin has the same law as the ground-state energy per spin), $g_{\mathrm{s}}$ denotes its typical (self-averaging) value, and the side $r>g_{\mathrm{s}}$ therefore corresponds, at $T=0$, to free energies below the typical value, i.e. the lower tail of the replica computation, which is the side generated by positive $n$ — equivalently, $\Phi(n,T)=f_{\mathrm{typ}}+c_5 n^5+o(n^5)$ for small positive $n$ below $T_c$, with $c_5=-(9/640)\beta^5[T\dot q(0)]^3$ expressed through the slope at $x=0$ of the Parisi function $q(x)$ (at $T=0$: $c_5\beta^{-5}\to-0.0057766$). This exponent and amplitude have since been proved — the exponent $6/5$ at positive and zero temperature by Aronow and Lopatto (arXiv:2609.09103), and the $q(x)$-dependent amplitude $c_\beta=(9/640)\beta^3\rho_\beta(0)^{-3}$ by Pei (arXiv:2610.06480).

2. Order of small deviations and validity of matching. Determine the true order of the typical fluctuations of the SK free energy (and ground-state energy) below $T_c$: is the matching prediction correct, i.e. deviations of the free energy per spin of order $N^{-5/6}$ (total free energy of order $N^{1/6}$, variance $\Theta(N^{1/3})$)? The competing $N^{-3/4}$ scaling is now excluded, since it would require a variance of the total free energy of order $N^{1/2}$, above the $N^{7/15}$ upper bound of Aronow and Lopatto (arXiv:2609.09103); the open question is therefore whether the matching prediction $N^{-5/6}$ — now bracketed rather than identified — is the exact order. This includes settling whether the smooth matching between the central peak and the large-deviation tail is valid below $T_c$.

3. Finite-temperature theory. Extend the rigorous large-deviation principle beyond zero temperature: establish the phase diagram of $\Phi(n,T)$ in the $(n,T)$ plane (including the de Almeida–Thouless line of the replicated system and the first-order transition at $n=0$), settle the non-commuting limits $n\to0$ and $N\to\infty$ (derivatives of $\Phi$ at $n=0$ are not obtainable by first taking $N\to\infty$), and quantitatively validate or refute the predicted finite-temperature $L(f)$ — including its $q(x)$-dependent amplitudes — against numerical data, which so far exist only at $T=0$.

These components serve one objective: a single coherent theory of the SK free-energy distribution in which the replica prediction, its rigorous derivation, and its numerical validation agree. The order of the fluctuations below $T_c$ (component 2) and the finite-temperature theory (component 3) are separate questions, and a complete account requires settling both. The finite-temperature theory is one program whose sub-goals (the $\Phi(n,T)$ phase diagram with its de Almeida–Thouless line and $n=0$ first-order transition, the non-commuting $n\to0$/$N\to\infty$ limits, and the finite-temperature numerical validation) are all parts of that program.

References

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