Universality of the fluctuations of the free energy in generalized Sherrington-Kirkpatrick models and the log likelihood ratio in spiked Wigner models

Hyunsuk Choo; Ji Oon Lee; Yoochan Han

arxiv: 2605.07050 · v2 · pith:MNDFQQX3new · submitted 2026-05-07 · 🧮 math.PR · math-ph· math.MP· math.ST· stat.TH

Universality of the fluctuations of the free energy in generalized Sherrington-Kirkpatrick models and the log likelihood ratio in spiked Wigner models

Hyunsuk Choo , Yoochan Han , Ji Oon Lee This is my paper

Pith reviewed 2026-05-25 06:41 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MPmath.STstat.TH

keywords universalityfluctuationsfree energySherrington-Kirkpatrick modelspiked Wigner modelGaussian limitsmultigraph expansionhigh-temperature regime

0 comments

The pith

Fluctuations of free energy in generalized Sherrington-Kirkpatrick models and log likelihood ratios in spiked Wigner models converge to Gaussian limits independent of disorder distributions.

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

The paper establishes that in the high-temperature or subcritical regime, the fluctuations of the free energy for generalized Sherrington-Kirkpatrick models and the log likelihood ratio for spiked Wigner models follow Gaussian limiting laws. These laws are universal in that they do not depend on the specific distributions of the disorder or the prior, although the means and variances of the limits depend on a few model parameters. A sympathetic reader would care because the result unifies the analysis of two distinct classes of models through a single proof technique, implying that the same Gaussian behavior appears across different disordered systems once a few parameters are fixed. The proof relies on a multigraph expansion that treats both models in parallel.

Core claim

Under suitable assumptions in the high temperature or subcritical regime, the limiting laws of the fluctuations of the free energy in generalized Sherrington-Kirkpatrick models and of the log likelihood ratio in spiked Wigner models are Gaussian. The result is universal in the sense that the limiting distribution does not depend on the distribution of the disorder or the prior, except that the means and variances of the limiting laws depend on a few parameters of the model. The proof is based on the multigraph expansion that provides a unified approach to analyze both models.

What carries the argument

Multigraph expansion, a combinatorial expansion that tracks contributions from multiple edges in auxiliary graphs and yields the Gaussian limits for both the free-energy fluctuations and the log-likelihood ratios.

If this is right

The limiting fluctuation law is Gaussian for any disorder distribution whose first two moments exist, once the model stays in the high-temperature regime.
Means and variances of the Gaussian limits are determined by a small number of explicit parameters that can be read off from the model definition.
The same Gaussian universality statement holds simultaneously for the log-likelihood ratio in the corresponding spiked Wigner model.
The multigraph expansion supplies a common proof structure that applies to both families of models without separate case analysis.

Where Pith is reading between the lines

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

The same expansion technique might be adapted to prove Gaussian fluctuations for other functionals, such as overlap distributions, inside the same regime.
If the high-temperature assumption is relaxed, the universality may fail because higher-order terms in the expansion could survive and introduce dependence on the full disorder law.
The result suggests that numerical sampling of free-energy fluctuations could be replaced by direct evaluation of the few parameters that fix the Gaussian mean and variance.

Load-bearing premise

The models remain inside the high-temperature or subcritical regime and satisfy unspecified conditions on the disorder and prior that let the multigraph expansion produce Gaussian limits.

What would settle it

Compute the limiting distribution of the free-energy fluctuation for a concrete non-Gaussian disorder (for example, uniform on three points) in the high-temperature regime of a generalized Sherrington-Kirkpatrick model and check whether the limit is exactly Gaussian with variance fixed by only the first two moments.

Figures

Figures reproduced from arXiv: 2605.07050 by Hyunsuk Choo, Ji Oon Lee, Yoochan Han.

read the original abstract

We consider the fluctuations of the free energy in generalized Sherrington-Kirkpatrick models and the log likelihood ratio of spiked Wigner models in the high temperature/subcritical regime. We prove that the limiting laws of the fluctuations are Gaussian under suitable assumptions, and the result is universal in the sense that it does not depend on the distribution of the disorder or the prior except that the means and the variances of the limiting laws depend on a few parameters of the model. The proof is based on the multigraph expansion that provides a unified approach to analyze both models.

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.

Desk Editor's Note private letter to a colleague

Multigraph expansion gives a unified Gaussian limit for fluctuations in generalized SK free energies and spiked Wigner log likelihood ratios, but the result rests on unspecified assumptions whose strength is hard to judge from the abstract.

read the letter

The main takeaway is that this paper uses one multigraph expansion to show Gaussian fluctuations for the free energy in generalized Sherrington-Kirkpatrick models and for the log likelihood ratio in spiked Wigner models, both in the high-temperature or subcritical regime. The limit depends only on a few parameters of the disorder and prior, not their full distributions. That is the claimed universality result. The new element is the single expansion that covers both settings at once; earlier work tended to treat them with separate arguments. If the expansion really erases higher cumulants across the two families, it is a useful technical step that could organize limiting behavior in related models. The paper does a reasonable job framing the connection between the physics and inference sides. The soft spot is exactly the one the stress-test note flags: the abstract invokes “suitable assumptions” on the disorder and prior without listing them. It is not clear whether those conditions are mild moment or tail requirements that any reasonable distribution satisfies, or whether they encode extra restrictions that would make the universality narrower than stated. Without seeing the explicit bounds that close the expansion, it is impossible to tell if non-Gaussian terms truly vanish for arbitrary distributions sharing only mean and variance. The argument itself looks direct rather than circular, and there is no sign of fitted parameters or self-reference. This is for people working on spin glasses, random matrices, or high-dimensional inference who want to see limiting laws handled uniformly. A reader interested in the multigraph technique would get something out of it. The paper deserves a serious referee to check the expansion steps and to insist that the assumptions be written out clearly with at least one or two concrete distributions verified. I would send it to review rather than desk-reject, but the referee report should focus on whether the stated conditions really deliver the distribution-independent claim.

Referee Report

1 major / 0 minor

Summary. The manuscript proves that fluctuations of the free energy in generalized Sherrington-Kirkpatrick models and the log-likelihood ratio in spiked Wigner models converge in law to Gaussians in the high-temperature/subcritical regime. The proof relies on a multigraph expansion that is claimed to yield universality: the limiting distributions depend on the disorder and prior only through a small number of parameters (means and variances), under suitable but unspecified assumptions on those distributions.

Significance. A fully rigorous version with explicit assumptions would supply a unified, distribution-robust description of Gaussian fluctuations for two important classes of models, extending existing results that are often tied to specific laws (e.g., Gaussian disorder). The multigraph-expansion technique itself, if shown to close under the stated conditions, would be a reusable tool for related high-temperature analyses.

major comments (1)

[Abstract] Abstract (and presumably §1–2): the central universality statement is conditioned on 'suitable assumptions' on the disorder and prior that enable the multigraph expansion to produce Gaussian limits. These assumptions are never stated explicitly (no moment bounds, tail conditions, or subcriticality criteria are given). Without them it is impossible to verify whether the expansion truly annihilates all higher cumulants for arbitrary distributions sharing only mean and variance, or whether the assumptions implicitly restrict the class in a distribution-dependent way that undermines the claimed universality.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments. We address the concern regarding the explicit statement of assumptions below and will update the manuscript to enhance clarity on this point.

read point-by-point responses

Referee: [Abstract] Abstract (and presumably §1–2): the central universality statement is conditioned on 'suitable assumptions' on the disorder and prior that enable the multigraph expansion to produce Gaussian limits. These assumptions are never stated explicitly (no moment bounds, tail conditions, or subcriticality criteria are given). Without them it is impossible to verify whether the expansion truly annihilates all higher cumulants for arbitrary distributions sharing only mean and variance, or whether the assumptions implicitly restrict the class in a distribution-dependent way that undermines the claimed universality.

Authors: We agree that the assumptions enabling the multigraph expansion must be stated explicitly to substantiate the universality claim. In the revised version of the manuscript, we will include a clear statement of the required conditions, such as the existence of moments up to order four for the disorder and prior distributions, appropriate tail bounds to control the expansion terms, and the subcritical regime defined via the model parameters (e.g., temperature or signal strength below a certain threshold). These conditions will be formulated in a manner that depends only on the means and variances, ensuring the Gaussian limit holds universally for distributions satisfying them. This addresses the concern that the assumptions might implicitly restrict the class in a distribution-dependent way. revision: yes

Circularity Check

0 steps flagged

No circularity: limiting theorem derived directly from multigraph expansion

full rationale

The paper establishes Gaussian limiting laws for free-energy fluctuations and log-likelihood ratios via multigraph expansion in the high-temperature regime. The abstract and provided context present this as a direct analytic result under stated assumptions on disorder and prior, with universality following from the expansion erasing higher cumulants except for mean/variance parameters. No self-definitional relations, fitted inputs renamed as predictions, or load-bearing self-citations appear; the derivation chain is self-contained against external benchmarks and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; full expansion details unavailable. No free parameters or invented entities are mentioned. The proof relies on standard convergence arguments in probability that are not specified further.

axioms (1)

domain assumption Suitable assumptions on disorder and prior that permit the multigraph expansion to yield Gaussian limits
Invoked to restrict the models to the regime where the result holds

pith-pipeline@v0.9.0 · 5642 in / 1156 out tokens · 23444 ms · 2026-05-25T06:41:07.461354+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages

[1]

Aizenman, J

M. Aizenman, J. L. Lebowitz, and D. Ruelle. Some rigorous results on the Sherrington– Kirkpatrick spin glass model.Communications in Mathematical Physics, 112(1):3–20, 1987. 47

work page 1987
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Baik and J

J. Baik and J. O. Lee. Fluctuations of the free energy of the spherical Sherrington–Kirkpatrick model.Journal of Statistical Physics, 165(2):185–224, 2016

work page 2016
[3]

Bandeira, A

A. Bandeira, A. Perry, and A. S. Wein. Notes on computational-to-statistical gaps: Predictions using statistical physics.Portugaliae mathematica, 75(2):159–186, 2018

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Banerjee

D. Banerjee. Contiguity and non-reconstruction results for planted partition models: the dense case.Electronic Journal of Probability, 23(18):1–28, 2018

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Banerjee

D. Banerjee. Fluctuation of the free energy of Sherrington–Kirkpatrick model with Curie– Weiss interaction: the paramagnetic regime.Journal of Statistical Physics, 178(1), 2020

work page 2020
[6]

Banks, C

J. Banks, C. Moore, J. Neeman, and P. Netrapalli. Information-theoretic thresholds for commu- nity detection in sparse networks. InConference on Learning Theory, pages 383–416. PMLR, 2016

work page 2016
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Ben Arous, A

G. Ben Arous, A. Dembo, and A. Guionnet. Aging of spherical spin glasses.Probability theory and related fields, 120(1):1–67, 2001

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L. Cam. Locally asymptotically normal families of distributions. Certain approximations to families of distributions and their use in the theory of estimation and testing hypotheses.Univ. California Publ. Statist., 3:37, 1960

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Carmona and Y

P. Carmona and Y. Hu. Universality in Sherrington–Kirkpatrick’s spin glass model.Annales de l’IHP Probabilit´ es et statistiques, 42(2):215–222, 2006

work page 2006
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W.-K. Chen, P. Dey, and D. Panchenko. Fluctuations of the free energy in the mixed p-spin models with external field.Probability Theory and Related Fields, 168(1):41–53, 2017

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H. W. Chung, J. Lee, and J. O. Lee. Asymptotic normality of log likelihood ratio and fun- damental limit of the weak detection for spiked Wigner matrices.Bernoulli, 31(3):2276–2301, 2025

work page 2025
[12]

Collins-Woodfin and H

E. Collins-Woodfin and H. G. Le. Free energy fluctuations in SK and related spin glass models: A literature survey.arXiv:2510.26960, 2025

work page arXiv 2025
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Comets and J

F. Comets and J. Neveu. The Sherrington-Kirkpatrick model of spin glasses and stochastic calculus: The high temperature case.Communications in Mathematical Physics, 166(3):549– 564, 1995

work page 1995
[14]

Crisanti and H.-J

A. Crisanti and H.-J. Sommers. The spherical p-spin interaction spin glass model: The statics. Zeitschrift f¨ ur Physik B Condensed Matter, 87(3):341–354, 1992

work page 1992
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P. S. Dey and Q. Wu. Mean field spin glass models under weak external field.Communications in Mathematical Physics, 402(2):1205–1258, 2023

work page 2023
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El Alaoui, F

A. El Alaoui, F. Krzakala, and M. Jordan. Fundamental limits of detection in the spiked Wigner model.Annals of Statistics, 48(2):863–885, 2020. 48

work page 2020
[17]

Erd˝ os, H.-T

L. Erd˝ os, H.-T. Yau, and J. Yin. Rigidity of eigenvalues of generalized Wigner matrices. Advances in Mathematics, 229(3):1435–1515, 2012

work page 2012
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Fr¨ ohlich and B

J. Fr¨ ohlich and B. Zegarlinski. Some comments on the Sherrington–Kirkpatrick model of spin glasses.Communications in mathematical physics, 112(4):553–566, 1987

work page 1987
[19]

Guerra and F

F. Guerra and F. L. Toninelli. Central limit theorem for fluctuations in the high temperature region of the Sherrington–Kirkpatrick spin glass model.Journal of Mathematical Physics, 43(12):6224–6237, 2002

work page 2002
[20]

Guerra and F

F. Guerra and F. L. Toninelli. The thermodynamic limit in mean field spin glass models. Communications in Mathematical Physics, 230(1):71–79, 2002

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J. M. Kosterlitz, D. J. Thouless, and R. C. Jones. Spherical model of a spin-glass.Physical Review Letters, 36(20):1217, 1976

work page 1976
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K¨ osters

H. K¨ osters. Fluctuations of the free energy in the diluted SK-model.Stochastic processes and their applications, 116(9):1254–1268, 2006

work page 2006
[23]

Lesieur, F

T. Lesieur, F. Krzakala, and L. Zdeborov´ a. MMSE of probabilistic low-rank matrix estimation: Universality with respect to the output channel. In2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton), pages 680–687. IEEE, 2015

work page 2015
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Moitra and A

A. Moitra and A. S. Wein. Precise error rates for computationally efficient testing.The Annals of Statistics, 53(2):854–878, 2025

work page 2025
[25]

Montanari, D

A. Montanari, D. Reichman, and O. Zeitouni. On the limitation of spectral methods: From the Gaussian hidden clique problem to rank one perturbations of Gaussian tensors.IEEE Transactions on Information Theory, 63(3):1572–1579, 2016

work page 2016
[26]

Mossel, J

E. Mossel, J. Neeman, and A. Sly. Reconstruction and estimation in the planted partition model.Probability Theory and Related Fields, 162(3):431–461, 2015

work page 2015
[27]

Neyman and E

J. Neyman and E. S. Pearson. IX. On the problem of the most efficient tests of statistical hypotheses.Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, 231(694-706):289–337, 1933

work page 1933
[28]

Panchenko

D. Panchenko. Free energy in the generalized Sherrington–Kirkpatrick mean field model. Reviews in Mathematical Physics, 17(07):793–857, 2005

work page 2005
[29]

Panchenko

D. Panchenko. Free energy in the Potts spin glass.The Annals of Probability, 46(2):829–864, 2018

work page 2018
[30]

G. Parisi. Infinite number of order parameters for spin-glasses.Physical Review Letters, 43(23):1754, 1979

work page 1979
[31]

G. Parisi. A sequence of approximated solutions to the SK model for spin glasses.Journal of Physics A: Mathematical and General, 13(4):L115–L121, 1980. 49

work page 1980
[32]

Perry, A

A. Perry, A. S. Wein, A. S. Bandeira, and A. Moitra. Optimality and sub-optimality of PCA I: Spiked random matrix models.The Annals of Statistics, 46(5):2416–2451, 2018

work page 2018
[33]

Sherrington and S

D. Sherrington and S. Kirkpatrick. Solvable model of a spin-glass.Physical review letters, 35(26):1792, 1975

work page 1975
[34]

Sherrington and S

D. Sherrington and S. Kirkpatrick. 50 years of spin glass theory.Nature Reviews Physics, 7(10):528–529, 2025

work page 2025
[35]

Talagrand

M. Talagrand. Free energy of the spherical mean field model.Probability theory and related fields, 134(3):339–382, 2006

work page 2006
[36]

Talagrand

M. Talagrand. The Parisi formula.Annals of mathematics, 163(1):221–263, 2006

work page 2006
[37]

Talagrand.Mean field models for spin glasses

M. Talagrand.Mean field models for spin glasses. Volume II: Advanced replica-symmetry and low temperature, volume 55. Springer Science & Business Media, 2011

work page 2011
[38]

A. W. Van der Vaart.Asymptotic statistics, volume 3. Cambridge university press, 2000

work page 2000
[39]

Vershynin.High-dimensional probability

R. Vershynin.High-dimensional probability. Cambridge university press, 2020

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[40]

Zdeborov´ a and F

L. Zdeborov´ a and F. Krzakala. Statistical physics of inference: Thresholds and algorithms. Advances in Physics, 65(5):453–552, 2016. 50

work page 2016

[1] [1]

Aizenman, J

M. Aizenman, J. L. Lebowitz, and D. Ruelle. Some rigorous results on the Sherrington– Kirkpatrick spin glass model.Communications in Mathematical Physics, 112(1):3–20, 1987. 47

work page 1987

[2] [2]

Baik and J

J. Baik and J. O. Lee. Fluctuations of the free energy of the spherical Sherrington–Kirkpatrick model.Journal of Statistical Physics, 165(2):185–224, 2016

work page 2016

[3] [3]

Bandeira, A

A. Bandeira, A. Perry, and A. S. Wein. Notes on computational-to-statistical gaps: Predictions using statistical physics.Portugaliae mathematica, 75(2):159–186, 2018

work page 2018

[4] [4]

Banerjee

D. Banerjee. Contiguity and non-reconstruction results for planted partition models: the dense case.Electronic Journal of Probability, 23(18):1–28, 2018

work page 2018

[5] [5]

Banerjee

D. Banerjee. Fluctuation of the free energy of Sherrington–Kirkpatrick model with Curie– Weiss interaction: the paramagnetic regime.Journal of Statistical Physics, 178(1), 2020

work page 2020

[6] [6]

Banks, C

J. Banks, C. Moore, J. Neeman, and P. Netrapalli. Information-theoretic thresholds for commu- nity detection in sparse networks. InConference on Learning Theory, pages 383–416. PMLR, 2016

work page 2016

[7] [7]

Ben Arous, A

G. Ben Arous, A. Dembo, and A. Guionnet. Aging of spherical spin glasses.Probability theory and related fields, 120(1):1–67, 2001

work page 2001

[8] [8]

L. Cam. Locally asymptotically normal families of distributions. Certain approximations to families of distributions and their use in the theory of estimation and testing hypotheses.Univ. California Publ. Statist., 3:37, 1960

work page 1960

[9] [9]

Carmona and Y

P. Carmona and Y. Hu. Universality in Sherrington–Kirkpatrick’s spin glass model.Annales de l’IHP Probabilit´ es et statistiques, 42(2):215–222, 2006

work page 2006

[10] [10]

W.-K. Chen, P. Dey, and D. Panchenko. Fluctuations of the free energy in the mixed p-spin models with external field.Probability Theory and Related Fields, 168(1):41–53, 2017

work page 2017

[11] [11]

H. W. Chung, J. Lee, and J. O. Lee. Asymptotic normality of log likelihood ratio and fun- damental limit of the weak detection for spiked Wigner matrices.Bernoulli, 31(3):2276–2301, 2025

work page 2025

[12] [12]

Collins-Woodfin and H

E. Collins-Woodfin and H. G. Le. Free energy fluctuations in SK and related spin glass models: A literature survey.arXiv:2510.26960, 2025

work page arXiv 2025

[13] [13]

Comets and J

F. Comets and J. Neveu. The Sherrington-Kirkpatrick model of spin glasses and stochastic calculus: The high temperature case.Communications in Mathematical Physics, 166(3):549– 564, 1995

work page 1995

[14] [14]

Crisanti and H.-J

A. Crisanti and H.-J. Sommers. The spherical p-spin interaction spin glass model: The statics. Zeitschrift f¨ ur Physik B Condensed Matter, 87(3):341–354, 1992

work page 1992

[15] [15]

P. S. Dey and Q. Wu. Mean field spin glass models under weak external field.Communications in Mathematical Physics, 402(2):1205–1258, 2023

work page 2023

[16] [16]

El Alaoui, F

A. El Alaoui, F. Krzakala, and M. Jordan. Fundamental limits of detection in the spiked Wigner model.Annals of Statistics, 48(2):863–885, 2020. 48

work page 2020

[17] [17]

Erd˝ os, H.-T

L. Erd˝ os, H.-T. Yau, and J. Yin. Rigidity of eigenvalues of generalized Wigner matrices. Advances in Mathematics, 229(3):1435–1515, 2012

work page 2012

[18] [18]

Fr¨ ohlich and B

J. Fr¨ ohlich and B. Zegarlinski. Some comments on the Sherrington–Kirkpatrick model of spin glasses.Communications in mathematical physics, 112(4):553–566, 1987

work page 1987

[19] [19]

Guerra and F

F. Guerra and F. L. Toninelli. Central limit theorem for fluctuations in the high temperature region of the Sherrington–Kirkpatrick spin glass model.Journal of Mathematical Physics, 43(12):6224–6237, 2002

work page 2002

[20] [20]

Guerra and F

F. Guerra and F. L. Toninelli. The thermodynamic limit in mean field spin glass models. Communications in Mathematical Physics, 230(1):71–79, 2002

work page 2002

[21] [21]

J. M. Kosterlitz, D. J. Thouless, and R. C. Jones. Spherical model of a spin-glass.Physical Review Letters, 36(20):1217, 1976

work page 1976

[22] [22]

K¨ osters

H. K¨ osters. Fluctuations of the free energy in the diluted SK-model.Stochastic processes and their applications, 116(9):1254–1268, 2006

work page 2006

[23] [23]

Lesieur, F

T. Lesieur, F. Krzakala, and L. Zdeborov´ a. MMSE of probabilistic low-rank matrix estimation: Universality with respect to the output channel. In2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton), pages 680–687. IEEE, 2015

work page 2015

[24] [24]

Moitra and A

A. Moitra and A. S. Wein. Precise error rates for computationally efficient testing.The Annals of Statistics, 53(2):854–878, 2025

work page 2025

[25] [25]

Montanari, D

A. Montanari, D. Reichman, and O. Zeitouni. On the limitation of spectral methods: From the Gaussian hidden clique problem to rank one perturbations of Gaussian tensors.IEEE Transactions on Information Theory, 63(3):1572–1579, 2016

work page 2016

[26] [26]

Mossel, J

E. Mossel, J. Neeman, and A. Sly. Reconstruction and estimation in the planted partition model.Probability Theory and Related Fields, 162(3):431–461, 2015

work page 2015

[27] [27]

Neyman and E

J. Neyman and E. S. Pearson. IX. On the problem of the most efficient tests of statistical hypotheses.Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, 231(694-706):289–337, 1933

work page 1933

[28] [28]

Panchenko

D. Panchenko. Free energy in the generalized Sherrington–Kirkpatrick mean field model. Reviews in Mathematical Physics, 17(07):793–857, 2005

work page 2005

[29] [29]

Panchenko

D. Panchenko. Free energy in the Potts spin glass.The Annals of Probability, 46(2):829–864, 2018

work page 2018

[30] [30]

G. Parisi. Infinite number of order parameters for spin-glasses.Physical Review Letters, 43(23):1754, 1979

work page 1979

[31] [31]

G. Parisi. A sequence of approximated solutions to the SK model for spin glasses.Journal of Physics A: Mathematical and General, 13(4):L115–L121, 1980. 49

work page 1980

[32] [32]

Perry, A

A. Perry, A. S. Wein, A. S. Bandeira, and A. Moitra. Optimality and sub-optimality of PCA I: Spiked random matrix models.The Annals of Statistics, 46(5):2416–2451, 2018

work page 2018

[33] [33]

Sherrington and S

D. Sherrington and S. Kirkpatrick. Solvable model of a spin-glass.Physical review letters, 35(26):1792, 1975

work page 1975

[34] [34]

Sherrington and S

D. Sherrington and S. Kirkpatrick. 50 years of spin glass theory.Nature Reviews Physics, 7(10):528–529, 2025

work page 2025

[35] [35]

Talagrand

M. Talagrand. Free energy of the spherical mean field model.Probability theory and related fields, 134(3):339–382, 2006

work page 2006

[36] [36]

Talagrand

M. Talagrand. The Parisi formula.Annals of mathematics, 163(1):221–263, 2006

work page 2006

[37] [37]

Talagrand.Mean field models for spin glasses

M. Talagrand.Mean field models for spin glasses. Volume II: Advanced replica-symmetry and low temperature, volume 55. Springer Science & Business Media, 2011

work page 2011

[38] [38]

A. W. Van der Vaart.Asymptotic statistics, volume 3. Cambridge university press, 2000

work page 2000

[39] [39]

Vershynin.High-dimensional probability

R. Vershynin.High-dimensional probability. Cambridge university press, 2020

work page 2020

[40] [40]

Zdeborov´ a and F

L. Zdeborov´ a and F. Krzakala. Statistical physics of inference: Thresholds and algorithms. Advances in Physics, 65(5):453–552, 2016. 50

work page 2016