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arxiv: 2606.28298 · v1 · pith:4QUNYF3Vnew · submitted 2026-06-26 · ❄️ cond-mat.dis-nn · cond-mat.stat-mech

Stationary point complexity via minimal supersymmetry breaking

Jaron Kent-Dobias This is my paper

Pith reviewed 2026-06-29 01:30 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn cond-mat.stat-mech
keywords stationary point complexitysupersymmetry breakingHessian determinantrandom landscapesperceptronSherrington-Kirkpatrick modelorder parametersspectral density
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The pith

A four-index supersymmetric representation computes the complexity of stationary points by incorporating the absolute value of the Hessian determinant through minimal spontaneous supersymmetry breaking.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a supersymmetric method to count stationary points in random landscapes that correctly includes the sign of the Hessian determinant. It starts from an elegant but often inaccurate two-index supersymmetric form and expands it to four indices, where the absolute value arises naturally from spontaneous supersymmetry breaking in one superspace direction. Assuming no further symmetry breaking occurs reduces the calculation to five order parameters generated by the algebra of the breaking operator. These parameters connect directly to the geometry and eigenvalue spectrum of the stationary points, with the breaking parameter itself equal to the density of Hessian eigenvalues at zero. The approach is demonstrated on the annealed complexity of the perceptron and Sherrington-Kirkpatrick models and extends to the quenched case by promoting the order parameters to replica matrices.

Core claim

We develop an expanded 4-index supersymmetric representation of the complexity problem which incorporates the absolute value naturally via spontaneous supersymmetry breaking along a particular superspace direction. Positing that no additional symmetry breaking occurs implies the reduction to five order parameters corresponding to elements of a superspace operator algebra generated by the spontaneously SUSY-breaking operator. We relate the order parameters to the geometry and spectra of stationary points, showing that the SUSY-breaking order parameter corresponds to the spectral density of the Hessian at zero eigenvalue. We give examples of this formalism applied to calculate the annealed com

What carries the argument

The 4-index supersymmetric representation with spontaneous supersymmetry breaking along one superspace direction, which generates an operator algebra reducing the problem to five order parameters linked to stationary-point geometry and Hessian spectra.

If this is right

  • The annealed complexity of stationary points is obtained for the perceptron model.
  • The annealed complexity of stationary points is obtained for the Sherrington-Kirkpatrick model.
  • The same five order parameters extend to quenched complexity by becoming replica matrices.
  • The SUSY-breaking order parameter equals the spectral density of the Hessian at zero eigenvalue.
  • The order parameters encode geometric properties of the stationary points in addition to their spectra.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method may apply to other mean-field disordered systems where the sign of a determinant controls the counting of critical points.
  • Numerical sampling of Hessians in finite-dimensional versions of these models could test whether the five-parameter reduction persists beyond the mean-field limit.
  • The operator-algebra structure might connect to existing supersymmetric treatments of spin glasses or neural-network loss surfaces.

Load-bearing premise

No additional symmetry breaking occurs beyond the minimal spontaneous breaking along one chosen superspace direction.

What would settle it

A direct count of stationary points in the perceptron or Sherrington-Kirkpatrick model that yields a complexity different from the five-order-parameter prediction would show the reduction does not hold.

Figures

Figures reproduced from arXiv: 2606.28298 by Jaron Kent-Dobias.

Figure 1
Figure 1. Figure 1: Total complexity and marginal complexity of underparameterized nonlinear least squares. Complexities as a function of energy for the nonlinear least squares model with 𝑓 (𝑞) = 𝑞 2 + 𝑞 3 , 𝛼 = 1.5, and 𝑉0 = 0. Total complexity of stationary points is shown in red, with the line format indicating whether most stationary points are optima (𝑄Ψ = 0) or saddles (𝑄Ψ > 0). Complexity of marginal optima is shown in… view at source ↗
Figure 2
Figure 2. Figure 2: Total complexity of overparameterized nonlinear least squares. Complexities as a function of energy for the nonlinear least squares model with 𝑓 (𝑞) = 𝑞 2 + 𝑞 3 , 𝑉0 = 0, and a variety of loads 𝛼. The horizontal line shows the maximum complexity at saturation in 𝛼 which matches the maximum complexity when it occurs at positive energy 𝐸. Solid lines depict regimes dominated by optima (𝑄Ψ = 0) and dashed lin… view at source ↗
Figure 3
Figure 3. Figure 3: Threshold energy of nonlinear least squares as a function of load and target. Threshold energy 𝐸th in nonlinear least squares as a function of load 𝛼 and target 𝑉0 for two different disorder covariances 𝑓 (𝑞) = 1 2 𝑞 𝑝 . When the threshold energy is predicted to fall below zero, stationary points with spectral weight on zero eigenvalue exist all the way to the ground state. where 𝜎 : R → R is the activatio… view at source ↗
Figure 4
Figure 4. Figure 4: Complexity curves for the negative margin spherical perceptron. Complexity as a function of energy density 𝐸 for the spherical perceptron with quadratic activation and margin 𝜅 = − 1 2 for several values of load 𝛼. Solid lines depict energies dominated by optima (𝑄Ψ = 0) and dashed lines depict energies dominated by saddles (𝑄Ψ > 0). Several distinct regimes are exhibited. When 𝛼 = 0.5, the complexity is m… view at source ↗
Figure 5
Figure 5. Figure 5: Maximum total and marginal complexity of the perceptron. Complexities for the spherical perceptron with activation 𝜎(𝑦) = 1 2 (𝜅 − 𝑦) 2Θ(𝜅 − 𝑦) and load 𝛼 = 𝑀/𝑁. Total complexity is divided into three phases. Below the dotted line the total complexity occurs at positive energy density with a susy-preserving solution. Between the dotted line and the dashed line the total complexity occurs at zero energy den… view at source ↗
Figure 6
Figure 6. Figure 6: Complexities of the Sherrington–Kirkpatrick model. Total and marginal complexity of the SK model as a function of temperature 𝑇 and energy density 𝐸. The dashed line in the total complexity shows the 𝑄˜ 1 = 0 solution, corresponding to the paramagnetic state, where the complexity is always zero. The marginal complexity is defined by the point where the spectral density of the stationary points has a pseudo… view at source ↗
read the original abstract

The statistics of stationary points are a powerful way to understand mean-field random landscapes, and the Kac--Rice formula is a general way to compute them. A longstanding technical barrier to these calculations is the presence of the absolute value of the determinant of the Hessian. Neglecting the absolute value produces an elegant 2-index supersymmetric representation of the problem, but is often incorrect. We develop an expanded 4-index supersymmetric representation of the complexity problem which incorporates the absolute value naturally via spontaneous supersymmetry breaking along a particular superspace direction. Positing that no additional symmetry breaking occurs implies the reduction to five order parameters corresponding to elements of a superspace operator algebra generated by the spontaneously SUSY-breaking operator. We relate the order parameters to the geometry and spectra of stationary points, showing that the SUSY-breaking order parameter corresponds to the spectral density of the Hessian at zero eigenvalue. We give examples of this formalism applied to calculate the annealed complexity of several models, including the perceptron and the Sherrington--Kirkpatrick model. The framework is naturally extended to quenched complexity, where each order parameter corresponds to a replica matrix.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper develops a 4-index supersymmetric representation of the Kac-Rice complexity problem that incorporates |det H| through spontaneous supersymmetry breaking along one superspace direction. Positing that no further symmetry breaking occurs reduces the problem to five order parameters (elements of the algebra generated by the breaking operator), which are mapped to geometric and spectral properties of stationary points, with the SUSY-breaking parameter identified as the Hessian eigenvalue density at zero. Explicit calculations are provided for the annealed complexity of the perceptron and Sherrington-Kirkpatrick model, and the framework is extended to quenched complexity via replica matrices.

Significance. If the minimality assumption is justified, the approach supplies a controlled supersymmetric route to the absolute determinant that has long obstructed exact complexity calculations, yielding a small set of order parameters with direct ties to Hessian spectra and geometry. The explicit annealed calculations for the perceptron and SK model, together with the replica extension for quenched cases, demonstrate concrete utility and could facilitate progress on mean-field landscape statistics beyond the cases where the absolute value can be neglected.

major comments (2)
  1. [Abstract] Abstract: The reduction from the 4-index representation to exactly five order parameters is obtained solely by positing that 'no additional symmetry breaking occurs beyond the minimal spontaneous breaking along a particular superspace direction.' No self-consistency argument, Ward-identity check, or explicit verification that the operator algebra does not enlarge is supplied, yet this positing is load-bearing for the claimed mapping of the SUSY-breaking order parameter to the spectral density of the Hessian at zero eigenvalue.
  2. [Formalism (implied by abstract description)] The construction of the 4-index supersymmetric representation is presented as naturally incorporating |det H| via spontaneous breaking, but the manuscript supplies no derivation or check against known results (e.g., the spherical p-spin or SK model complexities) that would confirm the breaking remains minimal once the representation is imposed.
minor comments (1)
  1. [Abstract] Abstract: The five order parameters are described as 'elements of a superspace operator algebra generated by the spontaneously SUSY-breaking operator,' but their explicit definitions or algebraic relations are not listed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful report and for recognizing the potential of the 4-index supersymmetric framework. We address the two major comments below, providing additional context on the minimality assumption while agreeing that further clarification would strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract] The reduction from the 4-index representation to exactly five order parameters is obtained solely by positing that 'no additional symmetry breaking occurs beyond the minimal spontaneous breaking along a particular superspace direction.' No self-consistency argument, Ward-identity check, or explicit verification that the operator algebra does not enlarge is supplied, yet this positing is load-bearing for the claimed mapping of the SUSY-breaking order parameter to the spectral density of the Hessian at zero eigenvalue.

    Authors: The minimality assumption is motivated by the fact that the absolute-value determinant is encoded by a single additional Grassmann direction; the resulting operator algebra generated by this breaking operator is closed under the superspace multiplications and differentiations appearing in the Kac-Rice integral, yielding precisely five independent bilinears. While the manuscript does not contain an explicit Ward-identity derivation, the mapping of the breaking parameter to the Hessian eigenvalue density at zero follows directly from the saddle-point equations once the algebra is truncated at this level. We will add a short appendix deriving the closure of the algebra and verifying that higher-order operators do not appear at the saddle point. revision: partial

  2. Referee: [Formalism (implied by abstract description)] The construction of the 4-index supersymmetric representation is presented as naturally incorporating |det H| via spontaneous breaking, but the manuscript supplies no derivation or check against known results (e.g., the spherical p-spin or SK model complexities) that would confirm the breaking remains minimal once the representation is imposed.

    Authors: The 4-index representation is obtained by augmenting the standard 2-index superspace with an extra pair of Grassmann coordinates whose bilinear is responsible for the sign of the determinant; spontaneous breaking along one superspace direction then generates the absolute value without further manual intervention. The explicit annealed calculations for the perceptron and SK model already serve as consistency checks, reproducing the expected geometric and spectral features. A direct comparison with the spherical p-spin complexity (where |det H| effects are known) can be performed within the same five-order-parameter saddle point; we will include this comparison and a step-by-step derivation of the representation from the Kac-Rice formula in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation rests on explicit assumption rather than self-referential reduction

full rationale

The paper states an explicit premise ('Positing that no additional symmetry breaking occurs implies the reduction to five order parameters') and then maps those parameters to independent geometric quantities such as the Hessian spectral density at zero. No equations are shown to reduce to each other by construction, no parameters are fitted on a subset and then relabeled as predictions, and no self-citations or imported uniqueness theorems appear in the supplied text. The framework is applied to concrete models (perceptron, SK), indicating content beyond the initial positing. This is the normal case of a paper whose central step is an assumption whose validity is separate from circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central reduction depends on the positing of minimal supersymmetry breaking only; no free parameters or invented entities are explicitly introduced in the abstract, though the five order parameters function as derived quantities.

axioms (1)
  • ad hoc to paper No additional symmetry breaking occurs beyond the minimal spontaneous supersymmetry breaking along a particular superspace direction.
    This positing is required to reduce the problem to five order parameters from the superspace operator algebra.

pith-pipeline@v0.9.1-grok · 5721 in / 1263 out tokens · 24627 ms · 2026-06-29T01:30:00.171455+00:00 · methodology

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Works this paper leans on

59 extracted references · 28 canonical work pages

  1. [1]

    Journal of Physics A: Mathematical and General , publisher =

    Kurchan, Jorge , title =. Journal of Physics A: Mathematical and General , publisher =. 1991 , month =. doi:10.1088/0305-4470/24/21/011 , issn =

    work page doi:10.1088/0305-4470/24/21/011 1991

  2. [2]

    Journal of Physics A: Mathematical and General , publisher =

    Parisi, G and Rizzo, T , title =. Journal of Physics A: Mathematical and General , publisher =. 2004 , month =

    2004

  3. [3]

    and Dean, David S

    Bray, Alan J. and Dean, David S. , title =. Physical Review Letters , publisher =. 2007 , month =

    2007

  4. [4]

    Journal of Statistical Mechanics: Theory and Experiment , publisher =

    Castellani, Tommaso and Cavagna, Andrea , title =. Journal of Statistical Mechanics: Theory and Experiment , publisher =. 2005 , month =

    2005

  5. [5]

    , title =

    Fyodorov, Yan V. , title =. Physical Review Letters , publisher =. 2004 , month =

    2004

  6. [6]

    Journal of Physics A: Mathematical and Theoretical , publisher =

    Urbani, Pierfrancesco , title =. Journal of Physics A: Mathematical and Theoretical , publisher =. 2023 , month =. doi:10.1088/1751-8121/acb742 , issn =

    work page doi:10.1088/1751-8121/acb742 2023

  7. [7]

    SciPost Physics , publisher =

    Kent-Dobias, Jaron , title =. SciPost Physics , publisher =. 2025 , month =. doi:10.21468/scipostphys.18.5.158 , issn =

    work page doi:10.21468/scipostphys.18.5.158 2025

  8. [8]

    Physical Review E , publisher =

    Kent-Dobias, Jaron , title =. Physical Review E , publisher =. 2024 , month =. doi:10.1103/physreve.110.064148 , issn =

    work page doi:10.1103/physreve.110.064148 2024

  9. [9]

    SciPost Physics , publisher =

    Franz, Silvio and Parisi, Giorgio and Sevelev, Maxime and Urbani, Pierfrancesco and Zamponi, Francesco , title =. SciPost Physics , publisher =. 2017 , month =. doi:10.21468/scipostphys.2.3.019 , issn =

    work page doi:10.21468/scipostphys.2.3.019 2017

  10. [10]

    Physical Review X , publisher =

    Folena, Giampaolo and Franz, Silvio and Ricci-Tersenghi, Federico , title =. Physical Review X , publisher =. 2020 , month =. doi:10.1103/PhysRevX.10.031045 , issue =

    work page doi:10.1103/physrevx.10.031045 2020

  11. [11]

    Physical Review E , publisher =

    Kent-Dobias, Jaron and Kurchan, Jorge , title =. Physical Review E , publisher =. 2023 , month =

    2023

  12. [12]

    and Bray, A

    Aspelmeier, T. and Bray, A. J. and Moore, M. A. , title =. Physical Review Letters , publisher =. 2004 , month =

    2004

  13. [13]

    Physical Review B , publisher =

    Müller, Markus and Leuzzi, Luca and Crisanti, Andrea , title =. Physical Review B , publisher =. 2006 , month =

    2006

  14. [14]

    Journal of Physics A: Mathematical and General , publisher =

    Rizzo, Tommaso , title =. Journal of Physics A: Mathematical and General , publisher =. 2005 , month =. doi:10.1088/0305-4470/38/15/005 , issn =

    work page doi:10.1088/0305-4470/38/15/005 2005

  15. [15]

    , title =

    Fyodorov, Yan V. , title =. Journal of Statistical Physics , publisher =. 2019 , month =

    2019

  16. [16]

    Fyodorov, Y. V. and Tublin, R. , title =. Acta Physica Polonica B , publisher =. 2020 , number =. doi:10.5506/aphyspolb.51.1663 , issn =

    work page doi:10.5506/aphyspolb.51.1663 2020

  17. [17]

    Journal of Physics A: Mathematical and Theoretical , publisher =

    Fyodorov, Yan V and Tublin, Rashel , title =. Journal of Physics A: Mathematical and Theoretical , publisher =. 2022 , month =. doi:10.1088/1751-8121/ac6d8e , issn =

    work page doi:10.1088/1751-8121/ac6d8e 2022

  18. [18]

    2022 , month =

    Tublin, Rashel , title =. 2022 , month =

    2022

  19. [19]

    SciPost Physics , publisher =

    Kamali, Persia Jana and Urbani, Pierfrancesco , title =. SciPost Physics , publisher =. 2023 , month =. doi:10.21468/scipostphys.15.5.219 , issn =

    work page doi:10.21468/scipostphys.15.5.219 2023

  20. [20]

    2023 , month =

    Kamali, Persia Jana and Urbani, Pierfrancesco , title =. 2023 , month =

    2023

  21. [21]

    2024 , month =

    Urbani, Pierfrancesco , title =. 2024 , month =

    2024

  22. [22]

    2024 , month =

    Montanari, Andrea and Subag, Eliran , title =. 2024 , month =

    2024

  23. [23]

    Journal of Statistical Mechanics: Theory and Experiment , publisher =

    Urbani, Pierfrancesco , title =. Journal of Statistical Mechanics: Theory and Experiment , publisher =. 2024 , month =. doi:10.1088/1742-5468/ad0635 , issn =

    work page doi:10.1088/1742-5468/ad0635 2024

  24. [24]

    Advances in Neural Information Processing Systems , editor =

    Montanari, Andrea and Urbani, Pierfrancesco , title =. Advances in Neural Information Processing Systems , editor =. 2025 , volume =

    2025

  25. [25]

    Journal of Physics A: Mathematical and General , publisher =

    Gardner, E and Derrida, B , title =. Journal of Physics A: Mathematical and General , publisher =. 1988 , month =. doi:10.1088/0305-4470/21/1/031 , issn =

    work page doi:10.1088/0305-4470/21/1/031 1988

  26. [26]

    Journal of Physics A: Mathematical and General , publisher =

    Gardner, E and Derrida, B , title =. Journal of Physics A: Mathematical and General , publisher =. 1989 , month =. doi:10.1088/0305-4470/22/12/004 , issn =

    work page doi:10.1088/0305-4470/22/12/004 1989

  27. [27]

    Journal of Physics A: Mathematical and General , publisher =

    Gardner, E , title =. Journal of Physics A: Mathematical and General , publisher =. 1988 , month =. doi:10.1088/0305-4470/21/1/030 , issn =

    work page doi:10.1088/0305-4470/21/1/030 1988

  28. [28]

    Physical Review X , publisher =

    Yamamura, Atsushi and Mabuchi, Hideo and Ganguli, Surya , title =. Physical Review X , publisher =. 2024 , month =. doi:10.1103/physrevx.14.031054 , issn =

    work page doi:10.1103/physrevx.14.031054 2024

  29. [29]

    Physical Review E , publisher =

    Ghimenti, Federico and Sriram, Adithya and Yamamura, Atsushi and Mabuchi, Hideo and Ganguli, Surya , title =. Physical Review E , publisher =. 2026 , month =. doi:10.1103/k73p-5k1w , issn =

    work page doi:10.1103/k73p-5k1w 2026

  30. [30]

    Journal of Physics C: Solid State Physics , publisher =

    Bray, A J and Moore, M A , title =. Journal of Physics C: Solid State Physics , publisher =. 1980 , month =

    1980

  31. [31]

    Physical Review Letters , publisher =

    Sherrington, David and Kirkpatrick, Scott , title =. Physical Review Letters , publisher =. 1975 , month =. doi:10.1103/physrevlett.35.1792 , issn =

    work page doi:10.1103/physrevlett.35.1792 1975

  32. [32]

    Thouless, D. J. and Anderson, P. W. and Palmer, R. G. , title =. Philosophical Magazine , publisher =. 1977 , month =. doi:10.1080/14786437708235992 , issn =

    work page doi:10.1080/14786437708235992 1977

  33. [33]

    and Moore, M

    Aspelmeier, T. and Moore, M. A. , title =. Physical Review E , publisher =. 2022 , month =. doi:10.1103/physreve.105.034138 , issn =

    work page doi:10.1103/physreve.105.034138 2022

  34. [34]

    Journal of Physics A: Mathematical and General , publisher =

    Cavagna, Andrea and Giardina, Irene and Parisi, Giorgio and Mézard, Marc , title =. Journal of Physics A: Mathematical and General , publisher =. 2003 , month =. doi:10.1088/0305-4470/36/5/301 , issn =

    work page doi:10.1088/0305-4470/36/5/301 2003

  35. [35]

    Physical Review E , publisher =

    Annibale, Alessia and Cavagna, Andrea and Giardina, Irene and Parisi, Giorgio , title =. Physical Review E , publisher =. 2003 , month =

    2003

  36. [36]

    Journal of Physics A: Mathematical and General , publisher =

    Annibale, Alessia and Cavagna, Andrea and Giardina, Irene and Parisi, Giorgio and Trevigne, Elisa , title =. Journal of Physics A: Mathematical and General , publisher =. 2003 , month =

    2003

  37. [37]

    Journal of Physics A: Mathematical and General , publisher =

    Annibale, Alessia and Gualdi, Giulia and Cavagna, Andrea , title =. Journal of Physics A: Mathematical and General , publisher =. 2004 , month =

    2004

  38. [38]

    Journal of Physics A: Mathematical and Theoretical , publisher =

    Ros, Valentina and Roy, Felix and Biroli, Giulio and Bunin, Guy , title =. Journal of Physics A: Mathematical and Theoretical , publisher =. 2023 , month =

    2023

  39. [39]

    , title =

    Kurchan, J. , title =. Journal de Physique I , publisher =. 1992 , month =

    1992

  40. [40]

    Journal of Statistical Mechanics: Theory and Experiment , publisher =

    Pacco, Alessandro and Rosso, Alberto and Ros, Valentina , title =. Journal of Statistical Mechanics: Theory and Experiment , publisher =. 2025 , month =. doi:10.1088/1742-5468/adbe40 , issn =

    work page doi:10.1088/1742-5468/adbe40 2025

  41. [41]

    Journal of Physics A: Mathematical and Theoretical , publisher =

    Lacroix-A-Chez-Toine, Bertrand and Fyodorov, Yan V , title =. Journal of Physics A: Mathematical and Theoretical , publisher =. 2022 , month =. doi:10.1088/1751-8121/ac564a , issn =

    work page doi:10.1088/1751-8121/ac564a 2022

  42. [42]

    and Pacco, Alessandro and Ros, Valentina and Urbani, Pierfrancesco , title =

    Fournier, Samantha J. and Pacco, Alessandro and Ros, Valentina and Urbani, Pierfrancesco , title =. Physical Review E , publisher =. 2026 , month =. doi:10.1103/62sm-m7lw , issn =

    work page doi:10.1103/62sm-m7lw 2026

  43. [43]

    , title =

    Kac, M. , title =. Bulletin of the American Mathematical Society , publisher =. 1943 , month =

    1943

  44. [44]

    Rice, S. O. , title =. American Journal of Mathematics , publisher =. 1939 , month =

    1939

  45. [45]

    Proceedings of The First Mathematical and Scientific Machine Learning Conference , editor =

    Maillard, Antoine and Ben Arous, Gérard and Biroli, Giulio , title =. Proceedings of The First Mathematical and Scientific Machine Learning Conference , editor =. 2020 , month =

    2020

  46. [46]

    Physical Review X , publisher =

    Ros, Valentina and Ben Arous, Gérard and Biroli, Giulio and Cammarota, Chiara , title =. Physical Review X , publisher =. 2019 , month =

    2019

  47. [47]

    and Sommers, H.-J

    Crisanti, A. and Sommers, H.-J. , title =. Zeitschrift für Physik B Condensed Matter , publisher =. 1992 , month =

    1992

  48. [48]

    Kirkpatrick, T. R. and Thirumalai, D. , title =. Physical Review B , publisher =. 1987 , month =

    1987

  49. [49]

    and Leuzzi, L

    Crisanti, A. and Leuzzi, L. , title =. Physical Review Letters , publisher =. 2004 , month =

    2004

  50. [50]

    and Leuzzi, L

    Crisanti, A. and Leuzzi, L. , title =. Physical Review B , publisher =. 2006 , month =

    2006

  51. [51]

    Journal of Physics A: Mathematical and General , publisher =

    Cavagna, Andrea and Giardina, Irene and Parisi, Giorgio , title =. Journal of Physics A: Mathematical and General , publisher =. 1997 , month =

    1997

  52. [52]

    , title =

    Rosenblatt, F. , title =. Psychological Review , publisher =. 1958 , number =. doi:10.1037/h0042519 , issn =

    work page doi:10.1037/h0042519 1958

  53. [53]

    2013 , month =

    Stojnic, Mihailo , title =. 2013 , month =. 1306.3980v1 , eprintclass =

    Pith/arXiv arXiv 2013

  54. [54]

    and Perugini, Gabriele and Pittorino, Fabrizio and Saglietti, Luca , title =

    Annesi, Brandon Livio and Lauditi, Clarissa and Lucibello, Carlo and Malatesta, Enrico M. and Perugini, Gabriele and Pittorino, Fabrizio and Saglietti, Luca , title =. Physical Review Letters , publisher =. 2023 , month =. doi:10.1103/physrevlett.131.227301 , issn =

    work page doi:10.1103/physrevlett.131.227301 2023

  55. [55]

    and Malatesta, Enrico M

    Annesi, Brandon L. and Malatesta, Enrico M. and Zamponi, Francesco , title =. SciPost Physics , publisher =. 2025 , month =. doi:10.21468/scipostphys.18.4.118 , issn =

    work page doi:10.21468/scipostphys.18.4.118 2025

  56. [56]

    Journal of Statistical Mechanics: Theory and Experiment , publisher =

    Kent-Dobias, Jaron , title =. Journal of Statistical Mechanics: Theory and Experiment , publisher =. 2026 , month =. doi:10.1088/1742-5468/ae4587 , issn =

    work page doi:10.1088/1742-5468/ae4587 2026

  57. [57]

    , title =

    Rieger, H. , title =. Physical Review B , publisher =. 1992 , month =

    1992

  58. [58]

    , title =

    DeWitt, Bryce S. , title =. 1992 , address =

    1992

  59. [59]

    and Moustakas, Aris L

    Tsironis, Theodoros G. and Moustakas, Aris L. , title =. Physical Review E , publisher =. 2025 , month =. doi:10.1103/3mbj-xkgk , issn =

    work page doi:10.1103/3mbj-xkgk 2025