Universality of the free-energy small-deviation distribution in full replica-symmetry-breaking mean-field spin glasses

Parisi and Rizzo (Phys. Rev. B 81, 094201, 2010) computed the positive (speed-$N^2$) large deviations of the free energy in mean-field spin glasses with full replica symmetry breaking (fRSB) and conjectured that the small sample-to-sample fluctuations of the free energy in the whole fRSB class follow one universal law: in the rescaled variable $\delta=(f-\bar f_N)N^{5/6}/\sqrt{T\dot q(0)}$ the distribution is $P(\delta)=\exp[-G(\delta)]$, with tails $G(\delta)\simeq a_-|\delta|^{6/5}$ as $\delta\to-\infty$ and $G(\delta)\simeq a_+\delta^{12/5}$ as $\delta\to+\infty$, and with $G$, $a_-$, $a_+$ independent of the model and of the temperature throughout the spin-glass phase. Only the two tail asymptotics were obtained (from a divergent replica loop expansion, resummed with about 15% uncertainty in the amplitudes); the full universal function has never been derived, and neither the $N^{1/6}$ fluctuation scale it implies nor its universality has been established — numerically the scale is still contested against a rival per-spin $N^{-3/4}$ (total $N^{1/4}$) law, and rigorously the best bounds are far from sharp. The problem is to determine $G$ in full, settling the anomalous fluctuation law of fRSB spin glasses together with the associated large-deviation structure: a speed-$N$ rate function for negative deviations behaving as $|\Delta f|^{6/5}$ near zero and a speed-$N^2$ rate function for positive deviations behaving as $(\Delta f)^{12/5}$ near zero.

Background

A spin glass is a system with quenched (frozen) random interactions whose low-temperature behavior is dominated by frustration, i.e. the impossibility of simultaneously satisfying all random interaction terms. The archetype is the Sherrington-Kirkpatrick (SK) model: $N$ Ising spins $\sigma_i=\pm1$ with couplings $J_{ij}$ that are independent Gaussians of variance $1/N$ connecting every pair. Each disorder realization ("sample") $J$ has free energy $F_J=-T\log\sum_\sigma e^{-H_J(\sigma)/T}$, and the free-energy density $f_J=F_J/N$ concentrates as $N\to\infty$. Beyond the typical value, the natural questions are the statistics of the residual sample-to-sample fluctuations of $F_J$ and of its exponentially unlikely large deviations. For random systems these are encoded in large-deviation rate functions, and two speeds arise: negative deviations of $f$ (free energy below typical) cost $\exp[-N\,L(\Delta f)]$, while positive deviations (above typical) are far rarer, costing $\exp[-N^2\,L_2(\Delta f)]$ — "very large deviations", because they require a collective fluctuation of all $O(N^2)$ couplings rather than of an extensive subset. For replica-symmetric (RS) mean-field models the answer is complete and governed by random matrix theory: the spherical 2-spin spin glass (a model of continuous spins on a sphere with quadratic random couplings) has its ground state given by the largest eigenvalue of a matrix from the Gaussian orthogonal ensemble (the standard ensemble of large real-symmetric random matrices with independent Gaussian entries), its free-energy fluctuations are of order $N^{-2/3}$ per spin and follow the Tracy-Widom law (the universal distribution of the normalized largest eigenvalue of large random matrices), with the below-typical tail governed by rate proportional to $|\Delta f|^{3/2}$ at speed $N$ and the above-typical tail by rate proportional to $(\Delta f)^3$ at speed $N^2$. The SK model's low-temperature phase is instead described by Parisi's full replica symmetry breaking (fRSB). In the replica formalism, disorder averages are computed through $\overline{Z^n}$ at integer replica number $n$ and analytically continued to $n\to0$; the exact saddle point in the glassy phase is a continuous hierarchy encoded in an order-parameter function $q(x)$, $x\in[0,1]$, whose slope at the origin, $\dot q(0)$, enters the fluctuation formulas (for SK, $T\dot q(0)\to0.743368$ as $T\to0$). The fRSB saddle point is marginal: its Hessian has a continuum of zero modes (the replicon sector), so the loop expansion of the replica field theory diverges at every order and only a resummation can yield finite answers. Exploiting precisely this structure, Parisi and Rizzo computed both large-deviation functions of the SK free energy in a trilogy of papers: the 2008 Physical Review Letters (PRL) paper obtained the negative-deviation rate $L(\Delta f)\propto|\Delta f|^{6/5}$ (from a nonanalytic $O(n^5)$ structure of the replicated free energy $\Phi(n)$), the 2009 Physical Review B (PRB) paper mapped the phase diagram of $\Phi(n,T)$, extended the computation to all temperatures and matched small and large deviations, and the 2010 PRB — the source of this problem — computed the positive-deviation rate $L_2(\Delta f)\propto(\Delta f)^{12/5}$ via a four-loop resummed expansion, consistent with zero-temperature ground-state data. Matching the two rate functions near $\Delta f=0$ with the far tails of the small-deviation distribution yields the 2010 conjecture at the heart of this problem: in any fRSB model the typical fluctuation of the total free energy is anomalously small, of order $N^{1/6}$ (per spin $N^{-5/6}$ — a phenomenon called superconcentration, far below the $N^{1/2}$ scale of naive concentration bounds), and the rescaled variable $\delta=(f-\bar f_N)N^{5/6}/\sqrt{T\dot q(0)}$ (with $\bar f_N$ the disorder-averaged free-energy density at size $N$) is distributed as $P(\delta)=\exp[-G(\delta)]$ with tails $G(\delta)\simeq a_-|\delta|^{6/5}$ ($\delta\to-\infty$, $a_-=1.366\ldots$) and $G(\delta)\simeq a_+\delta^{12/5}$ ($\delta\to+\infty$, $a_+=0.36\ldots$), where $G$ — analogously to the Tracy-Widom law for the RS class — should be one and the same universal function for every model in the class and every temperature below the critical one. The $N^{1/6}$ scale was first conjectured by Crisanti, Paladin, Sommers and Vulpiani (1992) from an early replica computation of $\Phi(n)$ and is supported by the finite-size-scaling arguments of Aspelmeier, Billoire, Marinari and Moore (2008); rival numerical works support a per-spin fluctuation of order $N^{-3/4}$ instead (a total scale $N^{1/4}$), and the two exponents are notoriously hard to separate numerically. What is missing is everything between the two tails: the full function $G$ has never been derived, computed, or measured; the universality across models and temperatures is an unproven "natural assumption"; the loop-expansion manipulations (the $n\to0$ limit after introducing a block structure of size $n$, the restriction to equal diagonal blocks of the fluctuation matrix, the continuation from the perturbative regime to the large-deviation regime) lack mathematical control; and the amplitudes carry a resummation uncertainty of about 15%. Rigorous results so far only bracket the claim: superconcentration upper bounds and lower bounds of order $N^{-1}$ per spin, $o(N^{-1/2})$ ground-state-energy fluctuations without field (Chen, Handschy and Lerman 2018), and — in a 2026 breakthrough by Chen, Guionnet, Ko, Lacroix-A-Chez-Toine and Mourrat — a first rigorous large-deviation principle for the ground-state energy of Ising p-spin models (speed $N$, explicit rate function, non-quadratic near its minimum exactly when no external field is present), which nevertheless leaves open even the predicted $(r-\mathrm{gs})^{6/5}$ behavior near the minimum and does not touch the $N^2$-speed side.

Problem Statement

Let $f_J$ be the free-energy density of a fully connected Ising spin glass whose low-temperature phase is described by a full replica-symmetry-breaking (fRSB) Parisi solution — the archetype being the Sherrington-Kirkpatrick model at temperature $T<T_c$, below the critical temperature of the spin-glass transition; equivalently, a model whose large-deviation functional reduces to the cubic action for replicon fluctuations analyzed by Parisi and Rizzo (2010) — and let $\dot q(0)$ be the slope at the origin of its Parisi order-parameter function. Parisi and Rizzo conjectured that the sample-to-sample fluctuations of $f_J$ are anomalously small and universal: defining the rescaled variable $\delta=(f_J-\bar f_N)N^{5/6}/\sqrt{T\dot q(0)}$ (with $\bar f_N$ the disorder-averaged free-energy density at size $N$), the distribution should converge to $P(\delta)=\exp[-G(\delta)]$ with a single function $G$ — the same for every model in the fRSB class and at every temperature in the spin-glass phase, including the $T\to0$ limit — whose tails behave as $G(\delta)\simeq a_-|\delta|^{6/5}$ as $\delta\to-\infty$ and $G(\delta)\simeq a_+\delta^{12/5}$ as $\delta\to+\infty$, with universal constants ($a_-=1.366\ldots$ known; $a_+=0.36\ldots$ currently carrying about 15% resummation uncertainty). Equivalently, on the large-deviation side: negative deviations of the free-energy density should occur with probability $\exp[-N\,L(\Delta f)]$ with $L(\Delta f)\propto|\Delta f|^{6/5}$ near zero, positive deviations with probability $\exp[-N^2\,L_2(\Delta f)]$ with $L_2(\Delta f)\propto(\Delta f)^{12/5}$ near zero, and the typical scale of total free-energy fluctuations should be $N^{1/6}$. The problem is to determine $G$ in full — resolving the fluctuation scale, in particular separating $N^{1/6}$ from the rival $N^{1/4}$ law, and resolving the shape of $G$, with controlled finite-size extrapolation and quantified uncertainties — and, as a separate requirement, to establish or refute the claimed model-independence and temperature-independence of $G$, the $N^{1/6}$ fluctuation scale, and the predicted rate-function exponents $6/5$ and $12/5$. Determining $G$ in a single fRSB model at a single temperature is necessary but not sufficient for this second requirement, which asks that the same $G$, scale, and exponents be shown (or refuted) to govern every model in the fRSB class and every temperature in the spin-glass phase. A refutation — for example a proof that two fRSB models, or one model at two temperatures, have different rescaled fluctuation laws, or that the fluctuation scale is not $N^{1/6}$ — equally solves the problem.

References

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