The rare-sample-dominated left tail of the disorder-averaged overlap distribution at the de Almeida-Thouless critical line of a mean-field spin glass in a field

At the critical point of the de Almeida-Thouless (AT) line of a mean-field Ising spin glass in a magnetic field, numerical simulations show that the disorder-averaged overlap distribution $P(q)$ develops an exponential left tail below the equilibrium overlap $q_0$ on the critical scaling scale $N^{1/3}(q-q_0)$, while the left tail of the sample-dependent distribution $P_J(q)$ of a typical sample is Gaussian or steeper. The averaged tail is generated entirely by a small, persistent fraction of atypical samples with bimodal $P_J(q)$ and a secondary peak far below $q_0$; consistently, the distribution over samples of the integrated overlap $X_J(q)$ at fixed averaged value develops a $1/X_J$ divergence at small $X_J$. Parisi and Ricci-Tersenghi (2012), who established this phenomenology on a degree-4 Bethe-lattice spin glass at field $H=0.7$, posed as open problems to derive the averaged left tail analytically by extending the Franz-Parisi-Virasoro constrained-replica technique from typical samples to the disorder average, and to identify the microscopic origin of the atypical samples, conjecturally a random effective field or temperature in the effective Hamiltonian. These problems remain open.

Background

A spin glass is a disordered magnetic system in which Ising spins ($\sigma_i=\pm1$) interact through random couplings $J_{ij}$, so that no periodic arrangement can satisfy all interactions and the low-temperature phase instead freezes into one of many random-looking equilibrium configurations. A fixed realization of the couplings is called a sample, and physical observables are averaged both over thermal fluctuations and over the disorder (the samples). The paradigmatic mean-field model is the Sherrington-Kirkpatrick (SK) model, in which all $N$ spins interact; diluted mean-field models replace the complete graph by a sparse random graph such as a random regular graph, where every spin has exactly $c$ neighbors (a Bethe lattice) and the exact solution is obtained by the cavity method, a recursive computation of the distribution of local fields acting on a spin added to a graph with one site removed (a cavity), implemented numerically by population dynamics.

The central order parameter is the overlap $q=N^{-1}\sum_i\sigma_i\tau_i$ between two independent equilibrium replicas, that is, two copies of the system with identical couplings evolving independently. For a given sample the thermal fluctuations of $q$ are described by the sample-dependent distribution $P_J(q)$, and the measured quantity is the disorder average $P(q)=\overline{P_J(q)}$, where the overline denotes the average over samples. Parisi's replica symmetry breaking (RSB) solution - the exact mean-field solution obtained by introducing $n$ copies (replicas) of the system and taking the $n\to0$ limit, with the permutation symmetry among replicas spontaneously broken in the spin-glass phase - predicts that $P(q)$ has support over a range of overlaps and is not self-averaging: $P_J(q)$ fluctuates from sample to sample even as $N\to\infty$. A long-standing question is whether a magnetic field destroys the spin-glass phase: in mean-field models there is a critical line in the temperature-field plane, the de Almeida-Thouless (AT) line, below which the replica-symmetric solution is unstable and RSB sets in. On this line the system is critical: the replicon, the soft eigen-direction of the replica fluctuation matrix whose vanishing eigenvalue signals the instability, is massless, and finite-size crossover is governed by the anomalous scale $N^{1/3}$, the same scale as at the zero-field critical point.

The tails of $P(q)$, the probability of large overlap fluctuations, carry information about the organization of the low-temperature phase. Franz, Parisi and Virasoro (1992) introduced a constrained-replica technique: two replicas are coupled by a term favoring a fixed overlap, and the free-energy cost of the constraint yields the probability of overlap fluctuations; Billoire, Franz and Marinari (2003) used it to derive the typical-sample tail $(1/N)\log P_N(q)\simeq -A(|q|-q_{EA})^3$ in the SK spin-glass phase, where $q_{EA}$ is the equilibrium (Edwards-Anderson) overlap. These computations describe typical samples and do not describe the disorder average when that average is dominated by rare, atypical samples. Sample-to-sample fluctuations are conveniently quantified by the integrated overlap $X_J(q)=\int_{-1}^{q}P_J(q')dq'$, the probability in sample $J$ of an overlap below $q$, and its disorder-averaged moments $X_k(q)=\overline{X_J(q)^k}$, with $X_1(q)=x(q)$ the averaged distribution itself; the limit $k\to0$, extracted from $\log X_k(q)/k$, suppresses the contribution of rare samples with large $X_J$ and therefore captures typical samples. Within RSB field theory, Mézard, Parisi, Sourlas, Toulouse and Virasoro (1984) and Parisi (1993, 2004), using stochastic stability (the invariance of the theory under small random perturbations of the couplings), predicted that in the thermodynamic-limit spin-glass phase the sample distribution of $X_J(q)$ has a power-law divergence $\Pi(s)\sim s^{X(q)-1}$ at the origin.

Parisi and Ricci-Tersenghi (2012) simulated the Ising spin glass on a degree-4 random regular graph with $J_{ij}=\pm1$ at field $H=0.7$, exactly at the AT critical point $T=T_c=0.73536$ with equilibrium overlap $q_0=0.67658$ (both computed to numerical precision by the cavity method), using parallel tempering - a Monte Carlo scheme in which copies of the system at different temperatures exchange configurations to accelerate equilibration - for sizes $N=2^6$ to $2^{14}$, of which only $N\le2^{12}$ were fully trusted (the two largest sizes suffered from limited sample counts and possible imperfect thermalization). They found that (i) the left tail of $P_J(q)$ of a typical sample is Gaussian or steeper, as the constrained-replica computation predicts; (ii) the disorder-averaged $P(q)$ nevertheless develops an exponential left tail in the scaling variable $N^{1/3}(q-q_0)$, which at not-too-large fields extends into the negative-$q$ region where typical samples have essentially no weight; (iii) this tail is generated entirely by a small fraction of atypical samples with bimodal $P_J(q)$ (a secondary peak far below $q_0$), a fraction that remains roughly constant as $N$ grows; and (iv) once the averaged value $X_1(q)$ falls below its value at the peak, $x(q_0)\simeq0.429$, the distribution over samples of $X_J(q)$ has its maximum at $X_J\simeq0$ and, for small $X_1$, develops a power-law divergence $1/X_J$ at small $X_J$, qualitatively echoing the RSB prediction. The same left-tail phenomenology in a field is documented in the SK model (Billoire and Coluzzi, 2003) and in diluted models relevant to finite dimensions (Leuzzi et al., 2009), and it propagates to three-dimensional simulations, where it motivated conditional analyses that track medians instead of averages because averages are dominated by a minority of samples (the 'silent majority' problem) and tail-robust susceptibility definitions.

The authors note the analogy with ferromagnets in a random field, where disorder-induced sample-to-sample fluctuations are strong and, as shown by Parisi and Sourlas (2002) for the random-field Ising model, the effect of the random field can be traded at the perturbative level for a dimensional reduction (critical behavior in $d$ dimensions mapping onto the pure system in $d-2$); they conjecture that the main source of the observed sample-to-sample fluctuations 'can be well described by a random temperature (or field) term in the effective Hamiltonian'. They state that deriving their results analytically by extending the Franz-Parisi-Virasoro technique from typical samples to the disorder average 'is a feasible task'. That derivation and the microscopic origin of the atypical samples are the subject of this problem.

Problem Statement

Consider a mean-field Ising spin glass in a constant magnetic field at the critical point of its de Almeida-Thouless line - concretely, the source simulations used a random regular graph of degree $c=4$ with $J_{ij}=\pm1$ couplings at field $H=0.7$, at $T=T_c=0.73536$ with equilibrium overlap $q_0=0.67658$; the same question can be posed for the SK model in a field or other mean-field variants, where the same tail phenomenology is documented. Provide a first-principles account of the left tail of the disorder-averaged overlap distribution $P(q)$ at this critical point and of the sample-to-sample fluctuations that generate it. This single objective comprises two main results:

(1) An analytic derivation of the disorder-averaged left tail for $q<q_0$ - equivalently of the averaged integrated overlap $x(q)=X_1(q)$, numerically an exponential in the scaling variable $N^{1/3}(q-q_0)$ with an extension into the negative-$q$ region at moderate fields - where the tail is dominated by rare atypical samples. The derivation should yield the tail shape and amplitude, its $N$-dependence and critical scaling window, and the decomposition of the average into typical- and rare-sample contributions.

(2) A microscopic characterization of the atypical samples: why a roughly $N$-independent fraction of samples has bimodal $P_J(q)$ with a secondary peak far below $q_0$, and a prediction for the distribution over samples of the integrated overlap $X_J(q)$ at fixed $X_1(q)$, including the observed $1/X_J$ divergence at small $X_J$. In particular, establish or quantitatively refute the conjecture that the dominant sample-to-sample fluctuations can be described by a random effective field or random temperature term in the effective Hamiltonian, in the spirit of the random-field Ising model correspondence of Parisi and Sourlas; if the conjecture fails, identify what does account for the fluctuations.

Larger, fully thermalized simulations of the same model at the same critical point - beyond the source data, which were restricted to $N\le2^{12}$ because of noise from the limited number of samples and possible imperfect thermalization at $N=2^{13}$-$2^{14}$ - testing the analytic predictions and the empirical scalings (the $N^{-1/3}$ collapse of the moments, the exponential averaged tail, the persistent atypical fraction, and the $1/X_J$ divergence) would provide additional confirmation of such a theory.

Completion boundary: the objective is met when the averaged left tail at the AT-line critical point follows from the microscopic model rather than from a fit, and the rare atypical samples that generate it are accounted for microscopically, with the source paper's random-field/random-temperature conjecture addressed.

References

1. Giorgio Parisi, Federico Ricci-Tersenghi, 'A numerical study of the overlap probability distribution and its sample-to-sample fluctuations in a mean-field model', Philosophical Magazine 92, 341-352 (2012). DOI: 10.1080/14786435.2011.634843; arXiv:1108.0759. https://doi.org/10.1080/14786435.2011.634843
2. S. Franz, G. Parisi, M. A. Virasoro, 'The replica method on and off equilibrium', Journal de Physique I 2, 1869-1880 (1992). DOI: 10.1051/jp1:1992115. https://doi.org/10.1051/jp1:1992115
3. Alain Billoire, Silvio Franz, Enzo Marinari, 'On the tail of the overlap probability distribution in the Sherrington Kirkpatrick model', Journal of Physics A: Mathematical and General 36, 15-27 (2003). DOI: 10.1088/0305-4470/36/1/302. https://doi.org/10.1088/0305-4470/36/1/302
4. Alain Billoire, Barbara Coluzzi, 'Magnetic field chaos in the Sherrington-Kirkpatrick model', Physical Review E 67, 036108 (2003). DOI: 10.1103/PhysRevE.67.036108. https://doi.org/10.1103/PhysRevE.67.036108
5. M. Mézard, G. Parisi, N. Sourlas, G. Toulouse, M. Virasoro, 'Nature of the Spin-Glass Phase', Physical Review Letters 52, 1156-1159 (1984). DOI: 10.1103/PhysRevLett.52.1156. https://doi.org/10.1103/PhysRevLett.52.1156
6. Giorgio Parisi, 'On the branching structure of the tree of states in spin glasses', Journal of Statistical Physics 72, 857-878 (1993). DOI: 10.1007/BF01048182. https://doi.org/10.1007/BF01048182
7. Giorgio Parisi, 'On the probability distribution of the overlap in spin glasses', International Journal of Modern Physics B 18, 733-743 (2004). DOI: 10.1142/S0217979204024355. https://doi.org/10.1142/S0217979204024355
8. Giorgio Parisi, Nicolas Sourlas, 'Scale invariance in disordered systems: the example of the random-field Ising model', Physical Review Letters 89, 257204 (2002). DOI: 10.1103/PhysRevLett.89.257204. https://doi.org/10.1103/PhysRevLett.89.257204
9. L. Leuzzi, G. Parisi, F. Ricci-Tersenghi, J. J. Ruiz-Lorenzo, 'Ising spin-glass transition in a magnetic field outside the limit of validity of mean-field theory', Physical Review Letters 103, 267201 (2009). DOI: 10.1103/PhysRevLett.103.267201. https://doi.org/10.1103/PhysRevLett.103.267201
10. R. A. Baños, A. Cruz, L. A. Fernandez, J. M. Gil-Narvion, A. Gordillo-Guerrero, M. Guidetti, D. Iñiguez, A. Maiorano, F. Mantovani, E. Marinari, V. Martin-Mayor, J. Monforte-Garcia, A. Muñoz Sudupe, D. Navarro, G. Parisi, S. Perez-Gaviro, F. Ricci-Tersenghi, J. J. Ruiz-Lorenzo, S. F. Schifano, B. Seoane, A. Tarancón, R. Tripiccione, D. Yllanes, 'Sample-to-sample fluctuations of the overlap distributions in the three-dimensional Edwards-Anderson spin glass', Physical Review B 84, 174209 (2011). DOI: 10.1103/PhysRevB.84.174209. https://doi.org/10.1103/PhysRevB.84.174209
11. M. Baity-Jesi, R. A. Baños, A. Cruz, L. A. Fernandez, J. M. Gil-Narvion, A. Gordillo-Guerrero, D. Iñiguez, A. Maiorano, F. Mantovani, E. Marinari, V. Martin-Mayor, J. Monforte-Garcia, A. Muñoz Sudupe, D. Navarro, G. Parisi, S. Perez-Gaviro, M. Pivanti, F. Ricci-Tersenghi, J. J. Ruiz-Lorenzo, S. F. Schifano, B. Seoane, A. Tarancon, R. Tripiccione, D. Yllanes, 'The three-dimensional Ising spin glass in an external magnetic field: the role of the silent majority', Journal of Statistical Mechanics: Theory and Experiment 2014, P05014 (2014). DOI: 10.1088/1742-5468/2014/05/P05014. https://doi.org/10.1088/1742-5468/2014/05/P05014
12. A. Billoire, A. Maiorano, E. Marinari, V. Martín-Mayor, D. Yllanes, 'Cumulative overlap distribution function in realistic spin glasses', Physical Review B 90, 094201 (2014). DOI: 10.1103/PhysRevB.90.094201. https://doi.org/10.1103/PhysRevB.90.094201
13. Matthew Wittmann, B. Yucesoy, Helmut G. Katzgraber, J. Machta, A. P. Young, 'Low-temperature behavior of the statistics of the overlap distribution in Ising spin-glass models', Physical Review B 90, 134419 (2014). DOI: 10.1103/PhysRevB.90.134419. https://doi.org/10.1103/PhysRevB.90.134419
14. Tommaso Rizzo, 'Replica trick for rare samples', Physical Review B 89, 174401 (2014). DOI: 10.1103/PhysRevB.89.174401. https://doi.org/10.1103/PhysRevB.89.174401
15. G. Parisi, F. Ricci-Tersenghi, T. Rizzo, 'Diluted mean-field spin-glass models at criticality', Journal of Statistical Mechanics: Theory and Experiment 2014, P04013 (2014). DOI: 10.1088/1742-5468/2014/04/P04013. https://doi.org/10.1088/1742-5468/2014/04/P04013
16. Cécile Monthus, Thomas Garel, 'Typical versus averaged overlap distribution in spin glasses: evidence for droplet scaling theory', Physical Review B 88, 134204 (2013). DOI: 10.1103/PhysRevB.88.134204. https://doi.org/10.1103/PhysRevB.88.134204
17. L. A. Fernandez, V. Martin-Mayor, G. Parisi, B. Seoane, 'Temperature chaos in 3D Ising spin glasses is driven by rare events', EPL 103, 67003 (2013). DOI: 10.1209/0295-5075/103/67003. https://doi.org/10.1209/0295-5075/103/67003
18. Michele Castellana, Giorgio Parisi, 'Non-perturbative effects in spin glasses', Scientific Reports 5, 8697 (2015). DOI: 10.1038/srep08697. https://doi.org/10.1038/srep08697
19. G. Parisi, F. Ricci-Tersenghi, D. Yllanes, 'Explicit generation of the branching tree of states in spin glasses', Journal of Statistical Mechanics: Theory and Experiment 2015, P05002 (2015). DOI: 10.1088/1742-5468/2015/05/P05002. https://doi.org/10.1088/1742-5468/2015/05/P05002
20. Maddalena Dilucca, Luca Leuzzi, Giorgio Parisi, Federico Ricci-Tersenghi, Juan J. Ruiz-Lorenzo, 'Spin glasses in a field show a phase transition varying the distance among real replicas (and how to exploit it to find the critical line in a field)', Entropy 22, 250 (2020). DOI: 10.3390/e22020250. https://doi.org/10.3390/e22020250
21. V. Martin-Mayor, J. J. Ruiz-Lorenzo, B. Seoane, A. P. Young, 'Numerical Simulations and Replica Symmetry Breaking' (book chapter), arXiv:2205.14089 (2022). https://arxiv.org/abs/2205.14089
22. L. H. Miranda-Filho, Yuliang Jin, 'Evidence of the de Almeida-Thouless transition in three-dimensional spin glasses', arXiv:2601.19192 (2026). https://arxiv.org/abs/2601.19192
23. Hang Du, Brice Huang, 'Overlap distribution of the critical Sherrington-Kirkpatrick model', arXiv:2608.08752 (2026). https://arxiv.org/abs/2608.08752
24. Silvio Franz, Giorgio Parisi, Federico Ricci-Tersenghi, 'Overlap locking and non-perturbative effects in spin glasses', arXiv:2602.18399 (2026). https://arxiv.org/abs/2602.18399