Rigorous status of the stochastic-stability hypotheses underlying the local fluctuation-dissipation relation and site thermometry in aging spin glasses

The open problem addressed here is the rigorous status of the unproved assumptions underlying the local fluctuation-dissipation relation derived in Giorgio Parisi's letter 'Local fluctuation dissipation relation' (EPL 65, 103-108 (2004), DOI: 10.1209/epl/i2003-10058-6; companion paper J. Phys. A 36, 10773 (2003)). The derivation rests on the following unproved assumptions: (i) intensive self averaging quantities of the M-clone ensemble are smooth functions of the inter-clone coupling strength $\epsilon$ as $M,N\to\infty$ and $\epsilon\to0$, in the dynamical sense made explicit in both papers: as $\epsilon\to0$, the local correlation functions and the response functions tend to their $\epsilon=0$ values uniformly in time; (ii) the limits $t\to\infty$ and $\delta\to 0$ may be exchanged when dynamical susceptibilities are matched to equilibrium moments of the local overlap distribution; (iii) local two-time quantities admit well-defined, non-oscillatory aging limits - assumed in the companion paper as the existence of $\lim_{t\to\infty}\chi_i^{(s)}(t)=\chi_i^{(s)}$ and of $X_i(C)=\lim_{t\to\infty}X_i^{(t)}(C)$, and, in inter-site form, as the Montanari-Ricci-Tersenghi transition-function hypothesis $C_i(t,t_{\mathrm w})=f_{ij}[C_j(t,t_{\mathrm w})]$ with continuous $f_{ij}$ in the $t,t_{\mathrm w}\to\infty$ limit; (iv) the conclusions are independent of the form of the inter-clone coupling $H_{\mathrm R}$ and of the order in which the limits $M\to\infty$, $N\to\infty$, $\epsilon\to 0$ are taken. From these assumptions the papers derive the local fluctuation-dissipation relation $-\mathrm dR_i/\mathrm dC_i = \beta x_i(C_i)$ (with $x_i$ the cumulative of the site-local overlap distribution) and the thermometric indistinguishability of sites. No proof or refutation of these assumptions is known in the literature for any concrete model class. Numerical tests support the local relation in coarsening (domain-growth) and one-step replica-symmetry-breaking (1RSB) models but are, in Montanari and Ricci-Tersenghi's words, 'more ambiguous (and probably very hard to settle numerically)' for the full replica-symmetry-breaking (FRSB) Viana-Bray case, where the probability distributions of the local fields computed with the SPT algorithm (a message-passing algorithm for single-site static quantities, the finite-temperature generalization of survey propagation) oscillate indefinitely; the rigorous Kac-limit analysis of Franz and Toninelli establishes convergence of local observables only for almost every perturbation and explicitly leaves continuity at $\epsilon\equiv 0$ open. The problem therefore remains open: establish for which classes of disordered or glassy models these assumptions can be rigorously justified, or determine the consequences (oscillatory aging, non-smooth $\epsilon$-dependence, site-dependent thermometry) when they fail.

Background

A spin glass is a magnet whose interactions are frozen-in random: the Hamiltonian is $H_J(\sigma)=-\sum_{ij} J_{ij}\sigma_i\sigma_j$ (Ising spins $\sigma_i=\pm1$, $i=1,\dots,N$) with the couplings $J_{ij}$ drawn once and for all from a fixed distribution ('quenched disorder', labelled by $J$). Standard theoretical laboratories are the fully connected Sherrington-Kirkpatrick (SK) model, the finite-dimensional Edwards-Anderson (EA) model (couplings only between nearest neighbours on a lattice), and the diluted Viana-Bray (VB) model (random sparse graph); 'Kac' models interpolate between these by letting the interaction range grow with system size. Below the freezing temperature these systems age: prepared by a quench, their observables keep evolving on timescales set by the waiting time $t_{\mathrm w}$ since the quench, and time-translation invariance is lost. At equilibrium, the fluctuation-dissipation theorem (FDT) fixes the response of an observable to a small conjugate field in terms of its autocorrelation, e.g. for the global correlation $C(t_w,t)=\langle\sigma(t_w)\sigma(t)\rangle_J$ and integrated response $R(t_w,t)$, FDT gives $-\mathrm dR/\mathrm dC=\beta$ ($\beta=1/T$). Out of equilibrium one instead observes a generalized fluctuation-dissipation relation (FDR): the parametric plot of $R$ against $C$ is a curve $-\mathrm dR/\mathrm dC=\beta X(C)$ with $0\le X(C)\le 1$, defining an effective temperature $T_{\mathrm{eff}}(C)=T/X(C)$. In mean-field models this was derived dynamically (Cugliandolo-Kurchan), and Franz, Mézard, Parisi and Peliti (1998) argued it holds generally in systems satisfying stochastic stability - the property that weak random perturbations of the Hamiltonian act smoothly on the quenched (fixed-sample) equilibrium state, established rigorously in mean field by Aizenman and Contucci (1998) for global equilibrium quantities. The static input is the order-parameter function of replica-symmetry breaking (RSB): for two independent equilibrium configurations $\sigma,\tau$ in the same $J$, the overlap $q(\sigma,\tau)=N^{-1}\sum_i\sigma_i\tau_i$ has a nontrivial sample-dependent distribution $P_J(q)$ (averaged over $J$: $P(q)$), and the Parisi function is its cumulative $x(q)=\int_0^q P(q')\,\mathrm dq'$. The static-dynamic dictionary then reads $X(C)=x(C)$: measuring the aging effective temperature amounts to measuring the equilibrium overlap distribution. The letter concerns the local, site-resolved version. Simulations of heterogeneous aging (Castillo, Chamon, Cugliandolo, Kennett 2002; Montanari and Ricci-Tersenghi 2003) found that the local correlation $C_i(t_{\mathrm w},t)=\langle\sigma_i(t_{\mathrm w})\sigma_i(t)\rangle_J$ and the local integrated response $R_i(t_{\mathrm w},t)$ (response at site $i$ to a field applied at site $i$ from time $t_{\mathrm w}$ on) obey a local FDR $-\mathrm dR_i/\mathrm dC_i=\beta X_i(C)$ whose function $X_i$ varies strongly from site to site, and, remarkably, that the effective temperature at a given time is the same at all sites - a thermometer coupled to any site reads the same value ('thermometric indistinguishability'). Parisi's derivation (EPL 2004; J. Phys. A 2003) introduces a clone construction to define a site-local overlap distribution: take $M$ identical copies ('clones') of the same sample with Hamiltonian $H_K(\sigma)=\sum_{a=1}^{M} H(\sigma^a)+\epsilon H_{\mathrm R}[\sigma]$, where $H_{\mathrm R}$ is a random inter-clone coupling (e.g. $H_{\mathrm R}=-\sum_{a,i}K_i^a\sigma_i^a\sigma_i^{a+1}$ with $K_i^a$ independent Gaussians), define the local overlap of two equilibrium configurations $\sigma,\tau$ of the clone ensemble as $q_i(\sigma,\tau)=M^{-1}\sum_{a=1}^{M}\sigma_i^a\tau_i^a$, and let $P_i(q)=\lim_{\epsilon\to0}\overline{P_i^\epsilon(q)}$ with the average $\overline{\cdot}$ over the couplings $K$ and with the limits $M\to\infty$, $N\to\infty$ taken before $\epsilon\to0$ (alternatively, per the letter, $\epsilon^2M,\epsilon^2N\gg1$; the companion paper gives instead $\epsilon M,\epsilon N\gg1$). The letter then computes multi-site susceptibilities to random perturbations $\Delta H_i^{(s)}$ with amplitude $\delta$ in two ways - as equilibrium moments of $P_i(q)$ and as time integrals over the aging dynamics - and matches them, obtaining moment relations such as $\chi_i^{(s)}=s\beta\int\mathrm dq\,x_i(q)\,q^{s-1}$ with $x_i(q)=\int_0^qP_i(q')\,\mathrm dq'$, hence the local FDR $X_i(C)=x_i(C)$; and, from positivity of the joint distribution $P(q_i,q_k)$ of local overlaps at two sites, $X_i(t)=X_k(t)$, i.e. thermometric indistinguishability. The derivation, however, rests on assumptions which the sources state and do not prove: (i) smoothness in $\epsilon$ - where 'intensive self average quantities' means per-site quantities whose $N\to\infty$ limit does not depend on the particular disorder realization ('Our central hypothesis is that all intensive self average quantities are smooth function of $\epsilon$ for small $\epsilon$... a kind of generalization of stochastic stability'), in the dynamical sense stated verbatim in both papers: as $\epsilon\to0$, the local correlation functions and the response functions tend to their $\epsilon=0$ values uniformly in time; (ii) commutativity of $t\to\infty$ with $\delta\to0$ ('the crucial hypothesis that the limits $t\to\infty$ and $\delta\to 0$ may be exchanged'); (iii) existence of well-defined, non-oscillatory aging limits for local two-time quantities - assumed in the companion paper through the existence assumptions $\lim_{t\to\infty}\chi_i^{(s)}(t)=\chi_i^{(s)}$ (eq. 51) and $X_i(C)=\lim_{t\to\infty}X_i^{(t)}(C)$ (eq. 58), and, in inter-site form, through the Montanari-Ricci-Tersenghi transition-function hypothesis: continuous functions $f_{ij},f_{ji}$ with $C_i(t,t_{\mathrm w})=f_{ij}[C_j(t,t_{\mathrm w})]$ and $C_j(t,t_{\mathrm w})=f_{ji}[C_i(t,t_{\mathrm w})]$ in the $t,t_{\mathrm w}\to\infty$ limit within a dynamically connected component, a condition they identify as necessary for the statics-dynamics connection to hold at the level of local (single-spin) observables; and (iv) independence of the form of $H_{\mathrm R}$ and of the order of the limits $M\to\infty$, $N\to\infty$, $\epsilon\to0$ ('The way in which we have coupled together our $M$ clones is irrelevant', companion paper, sec. 5.2). Assumption (i) is delicate in a way that rigorous work makes explicit: Franz and Toninelli (2004) proved for Kac-range models that free energy and local observables converge to their mean-field limits, but only for almost every realization of a perturbing randomness $\{\epsilon\}$, noting that Gibbs averages need not be continuous in the perturbation at $\epsilon\equiv0$ - exactly the point where assumption (i) would be needed. Later physics programs touch the same structures without settling them: Kurchan's multithermalization program (2023) - the proposal that an aging system with widely separated timescales behaves as if thermalized at a ladder of effective temperatures, one per timescale - argues at whole-system level that the Parisi scheme and slow dynamics imply each other, still assuming stochastic stability, and Franz, Parisi and Ricci-Tersenghi (2026) study 'overlap locking' - how two weakly coupled glassy systems synchronize their order parameters at intermediate coupling $1\ll\Delta H\ll N$ - which bears on how coupling details can matter (assumption (iv)) but is an equilibrium, two-system setting, not the aging clone ensemble.

Problem Statement

The problem is the rigorous status of the unproved assumptions on which Parisi's derivation of the local fluctuation-dissipation relation rests. Fix a class of disordered spin systems with quenched disorder $J$ (candidate classes: SK; Viana-Bray; Edwards-Anderson in $d=3$; Kac-range versions of EA), and for a fixed sample let $C_i(t_{\mathrm w},t)$ and $R_i(t_{\mathrm w},t)$ denote the site-local autocorrelation and integrated response in the aging regime after a quench, under the Langevin-type time evolution assumed in the sources. Define the clone ensemble $H_K(\sigma)=\sum_{a=1}^{M}H(\sigma^a)+\epsilon H_{\mathrm R}[\sigma]$ with random inter-clone coupling $H_{\mathrm R}$, the local overlap $q_i(\sigma,\tau)=M^{-1}\sum_{a=1}^{M}\sigma_i^a\tau_i^a$, and the local overlap distribution $P_i(q)=\lim_{\epsilon\to0}\overline{P_i^\epsilon(q)}$ for the fixed sample $J$, with the average $\overline{\cdot}$ over the inter-clone couplings $K$ and the limits $M\to\infty$ and $N\to\infty$ taken before $\epsilon\to0$ (alternatively, per the letter, $\epsilon^2M,\epsilon^2N\gg1$; per the companion paper, $\epsilon M,\epsilon N\gg1$); set $x_i(q)=\int_0^qP_i(q')\,\mathrm dq'$. The problem, in two branches. (A) Prove, for an explicit model class, conditions sufficient to derive the local FDR $-\frac{\mathrm dR_i}{\mathrm dC_i}=\beta\,x_i(C_i)$ (with $\beta=1/T$ the inverse bath temperature; $X_i$ defined by $-\mathrm dR_i/\mathrm dC_i=\beta X_i(C_i)$, the equal-time slope being $-\mathrm dR_i(t_{\mathrm w},t)/\mathrm dC_i(t_{\mathrm w},t)=\beta X_i(t_{\mathrm w},t)$, and the target relation asserted for the aging-limit function $X_i(C)=\lim_{t\to\infty}X_i^{(t)}(C)$ as in the sources - justifying the intervening limit exchanges among $N,M\to\infty$, $t_{\mathrm w},t\to\infty$ and $\epsilon,\delta\to0$ is part of the task), together with the thermometric indistinguishability $X_i(t)=X_k(t)$ that the sources derive from the same assumptions, for the site pairs to which that derivation applies ('two far away sites' in the sources' wording; in the fully connected and diluted classes, where no metric exists, any two distinct sites $i\ne k$), where the assumptions are: (i) every intensive (per-site) self-averaging quantity (one whose thermodynamic-limit value does not depend on the disorder realization) of the clone ensemble is a smooth function of $\epsilon$ for small $\epsilon$ (a site-local generalization of stochastic stability), in the sources' stated sense that as $\epsilon\to0$ the dynamical local correlation functions and the response functions tend to their $\epsilon=0$ values uniformly in time; (ii) the limits $t\to\infty$ and $\delta\to0$ commute when matching aging susceptibilities to equilibrium moments of $P_i(q)$, i.e. $\lim_{t\to\infty}\chi_i^{(s)}(t)=\chi_i^{(s)}=s\beta\int\mathrm dq\,x_i(q)\,q^{s-1}$, where $\chi_i^{(s)}(t)$ is the local susceptibility to the random perturbations $\Delta H_i^{(s)}$ of amplitude $\delta$ (under Langevin evolution, e.g. $\chi_i^{(2)}(t)=2\int_{t_{\mathrm w}}^{t}\mathrm dt'\,C_i(t',t)\,\partial R_i(t',t)/\partial t'$), with the sources' normalizations: variance $\delta$ for degree-1 local perturbations, $\delta/M$ for degree-2 local perturbations, and $\delta/M^4$ for the bilocal two-site perturbations used to probe $P(q_i,q_k)$; (iii) local two-time quantities admit well-defined non-oscillatory aging limits, i.e. the companion paper's existence assumptions $\lim_{t\to\infty}\chi_i^{(s)}(t)=\chi_i^{(s)}$ and $X_i(C)=\lim_{t\to\infty}X_i^{(t)}(C)$ hold and, in inter-site form, the Montanari-Ricci-Tersenghi transition functions converge: continuous functions $f_{ij},f_{ji}$ exist with $C_i(t,t_{\mathrm w})=f_{ij}[C_j(t,t_{\mathrm w})]$ and $C_j(t,t_{\mathrm w})=f_{ji}[C_i(t,t_{\mathrm w})]$ in the $t,t_{\mathrm w}\to\infty$ limit within a dynamically connected component, so that persistent temporal oscillations of local correlators are excluded; (iv) the derived quantities are independent of the choice of $H_{\mathrm R}$ and of the order in which $M\to\infty$, $N\to\infty$, $\epsilon\to0$ are taken. (B) Exhibit a concrete disordered or glassy model in the aging regime, to which the clone construction and the local $C_i,R_i$ apply, in which one of these assumptions fails: e.g. an intensive self-averaging quantity of the clone ensemble that is not continuous at $\epsilon\equiv0$; non-commutativity of $t\to\infty$ with $\delta\to0$ for an explicit susceptibility; persistent oscillatory aging of $C_i$ defeating the convergence of the inter-site transition functions; or a dependence of the resulting $X_i$ on the form of $H_{\mathrm R}$ - and determine the consequence of that failure for the derivation of the local FDR. A partial result counts toward branch (A) only insofar as the proven conditions are sufficient, in a stated model class and limit regime, to derive $-\mathrm dR_i/\mathrm dC_i=\beta x_i(C_i)$ with the derivation's limit exchanges justified. The global relation $X(C)=x(C)$ (Franz-Mézard-Parisi-Peliti) is the template whose proof the letter copies 'mutatis mutandis' at the site-resolved level; it is a consistency input to, not a consequence of, the local derivation.

References

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