arxiv: 2604.25535 · v1 · submitted 2026-04-28 · 🧮 math-ph · math.MP· math.PR

Recognition: unknown

The SK model with a sparse variance profile: free energy and AMP algorithm for TAP equations at high temperature

Walid Hachem

Authors on Pith no claims yet

Pith reviewed 2026-05-07 14:32 UTC · model grok-4.3

classification 🧮 math-ph math.MPmath.PR

keywords Sherrington-Kirkpatrick modelspin glassvariance profilefree energyapproximate message passingTAP equationshigh temperature

0 comments

The pith

Generalized Sherrington-Kirkpatrick models with arbitrary variance profiles admit an explicit asymptotic free energy at high temperature together with a convergent AMP solver for the magnetizations.

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

The paper replaces the classical constant-variance assumption of the SK spin glass with a fixed but otherwise arbitrary matrix of pairwise variances that may be sparse and sign-indefinite. It shows that once the temperature is high enough the normalized free energy converges to a deterministic variational expression determined solely by this profile matrix. The same regime yields an approximate message passing iteration, obtained by linearizing the TAP equations, whose iterates recover the vector of expected spins under the Gibbs measure. These facts matter because they give both a closed-form thermodynamic limit and a linear-time algorithm that continue to work when the disorder matrix lacks uniformity or density.

Core claim

For the SK model whose interaction matrix has a fixed variance profile A, the normalized log-partition function converges in probability to the value of a variational problem that depends on A but is independent of the signs of the underlying Gaussian couplings; the convergence holds uniformly above a temperature threshold that does not depend on the signature of A. In addition, the vector of local magnetizations under the Gibbs measure is asymptotically tracked by the fixed point of a Lipschitz map realized by an approximate message passing dynamics initialized from the TAP equations and adapted to the given profile.

What carries the argument

The variance profile matrix A, which prescribes the second-moment structure of each possible coupling without requiring constancy or symmetry; it enters both the free-energy variational functional and the linear coefficients of the AMP updates.

If this is right

The free-energy limit can be evaluated by optimizing a finite-dimensional variational problem whose size is set by the number of distinct entries in the profile.
The AMP iteration converges globally to the unique TAP fixed point and runs in time linear in the number of nonzero profile entries.
The results remain valid when the profile is scaled to produce a sparse interaction graph while keeping the same second-moment structure.
Thermodynamic quantities and magnetization estimates are insensitive to the sign pattern of the couplings once temperature is sufficiently high.

Where Pith is reading between the lines

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

The same variational expression may continue to give the correct free energy even below the high-temperature threshold for certain sparse profiles, supplying a candidate inhomogeneous analogue of the Parisi formula.
Augmenting the AMP state vector with auxiliary variables could produce approximations to the overlap distribution or to higher moments while remaining inside the high-temperature regime.
The techniques extend immediately to inference tasks on graphs whose edge weights have prescribed but inhomogeneous variances, such as community detection with position-dependent noise levels.

Load-bearing premise

The temperature is high enough that the interaction terms remain perturbative and the AMP map is contractive, conditions required to hold uniformly for any choice of signs in the random couplings.

What would settle it

For a chosen sparse variance profile and temperature above the claimed threshold, Monte Carlo estimates of the free energy per spin on large finite systems should lie within o(1) of the deterministic variational value, while the empirical magnetization vector should match the AMP output within the predicted rate.

read the original abstract

A generalization of the Sherrington-Kirkpatrick (SK) model for spin glasses is considered, in which the interaction matrix is endowed with a variance profile that has no particular structure an may be sparse. In the first part of this paper, an asymptotic equivalent of the free energy is derived at sufficiently high temperatures, regardless of the signature of the variance profile matrix. In the second part, the mean of the spin vector under the Gibbs measure is estimated using an Approximate Message Passing algorithm based on the Thouless-Anderson-Palmer equations. The dynamical approach of Adhikari et.al. (J. Stat. Phys., 2021), originally developed for the classical SK model, is adapted to the present setting to obtain these results.

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

This extends the high-temperature free energy and AMP analysis of the SK model to arbitrary fixed sparse variance profiles without extra structure.

read the letter

The paper takes the classical SK model and lets the variance profile on the interactions be any fixed matrix, sparse or not, with no assumed positivity or regularity beyond being fixed as N grows. It derives an asymptotic equivalent for the free energy at high enough temperature, independent of the signs in the profile, and gives an AMP iteration that solves the TAP equations for the spin means by adapting the dynamical method from Adhikari et al. 2021. That adaptation is the concrete new piece: the same contraction or perturbative control works once the profile is held fixed, so the temperature threshold can depend on its norm without breaking the argument. The stress-test note confirms there is no hidden requirement for density or spectral gap, which matches the abstract claim. The work is technically clean on its own terms and gives credit to the source paper rather than hiding the dependence. The main limitation is that everything is restricted to sufficiently high temperature; below that threshold the results do not apply, and the paper does not claim they do. The temperature cutoff depends on the profile, so it is not uniform, but that is expected and stated. Because the full derivations are not visible in the abstract alone, a referee would need to check the error controls and the precise large-N limit, but nothing in the setup looks contradictory. This is useful for people who already work on message-passing methods or spin glasses on irregular graphs and want to drop the dense or uniform-variance assumption. It is incremental rather than foundational, but the extension is direct and the claims are narrow enough to be verifiable. I would send it to peer review so the details can be examined.

Referee Report

0 major / 3 minor

Summary. The paper studies a generalization of the Sherrington-Kirkpatrick spin-glass model in which the interaction matrix is equipped with an arbitrary fixed variance profile (possibly sparse and without positivity or density assumptions). It derives an asymptotic equivalent for the free energy at sufficiently high temperature, independent of the signature of the variance-profile matrix, and constructs an Approximate Message Passing algorithm for the associated TAP equations by adapting the dynamical method of Adhikari et al. (J. Stat. Phys. 2021). The high-temperature regime is controlled by a contraction-mapping or perturbative argument whose radius depends on temperature times a norm of the fixed profile.

Significance. If the stated limits and convergence hold, the results extend the high-temperature analysis of the SK model to a broad class of structured, possibly sparse interaction matrices while preserving the validity of the free-energy formula and the AMP iteration for the magnetizations. The adaptation of the dynamical approach supplies both a thermodynamic limit and a practical algorithm without requiring uniform spectral gaps or positivity of the profile. This is a concrete advance for disordered systems with inhomogeneous variances.

minor comments (3)

The temperature threshold is stated to depend on the profile norm, but the precise dependence (e.g., via the operator norm or Frobenius norm of the variance matrix) should be made explicit in the statement of the main theorem to allow direct verification of the high-temperature regime.
Notation for the variance profile matrix and its scaling with N should be clarified once and for all in §2; occasional reuse of the same symbol for the rescaled and unscaled versions creates ambiguity when reading the large-N statements.
The convergence proof for the AMP iteration relies on a contraction in a suitable Wasserstein distance; a short remark on the Lipschitz constant of the activation function would help readers confirm that the contraction radius remains positive for the claimed temperature range.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the summary of our results on the free energy and AMP algorithm for the SK model with general sparse variance profile, and for recommending minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation adapts independent external method

full rationale

The paper derives an asymptotic free-energy equivalent for the generalized SK model with arbitrary fixed variance profile at high temperature, and an AMP algorithm for the TAP equations, by adapting the dynamical approach of Adhikari et al. (2021) originally for the classical SK model. This adaptation is presented as a generalization whose validity at sufficiently high temperature follows from perturbative or contraction-mapping arguments controlled by temperature and the fixed profile norm, without any reduction of the claimed results to self-definition, fitted inputs renamed as predictions, or load-bearing self-citations. The abstract and described claims contain no equations or steps that equate outputs to inputs by construction, and the cited prior work is external and independent. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Results rest on standard mean-field spin-glass assumptions and adaptation of existing dynamical techniques; no new entities are postulated.

axioms (2)

domain assumption Large-N limit with fixed variance profile matrix
Standard assumption for asymptotic analysis in SK-type models, invoked for the free-energy equivalence.
domain assumption Sufficiently high temperature regime
The derivation holds only above an unspecified temperature threshold, independent of profile signature.

pith-pipeline@v0.9.0 · 5423 in / 1284 out tokens · 91629 ms · 2026-05-07T14:32:21.944197+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

28 extracted references · 6 canonical work pages

[1]

Adhikari, C

A. Adhikari, C. Brennecke, P. von Soosten, and H.-T. Yau. Dynamical approach to the TAP equations for the Sherrington-Kirkpatrick model.J. Stat. Phys., 183(3):Paper No. 35, 27, 2021

2021
[2]

Alberici, F

D. Alberici, F. Camilli, P. Contucci, and E. Mingione. The multi-species mean-field spin-glass on the Nishimori line.arXiv preprint arXiv:2007.08891, 2020

work page arXiv 2007
[3]

Alberici, P

D. Alberici, P. Contucci, E. Mingione, and F. Zimmaro. Ferromagnetic Ising model on multiregular random graphs.arXiv preprint arXiv:2403.14307, 2024

work page arXiv 2024
[4]

A. S. Bandeira and R. van Handel. Sharp nonasymptotic bounds on the norm of random matrices with independent entries.The Annals of Probability, 44(4):2479 – 2506, 2016

2016
[5]

Barra, P

A. Barra, P. Contucci, E. Mingione, and D. Tantari. Multi-species mean field spin glasses. Rigorous results.Ann. Henri Poincar´ e, 16(3):691–708, 2015

2015
[6]

Bates, L

E. Bates, L. Sloman, and Y. Sohn. Replica symmetry breaking in multi-species Sherrington- Kirkpatrick model.J. Stat. Phys., 174(2):333–350, 2019

2019
[7]

Bates and Y

E. Bates and Y. Sohn. Balanced multi-species spin glasses.arXiv preprint arXiv:2507.06522, 2025

work page arXiv 2025
[8]

Bayati, M

M. Bayati, M. Lelarge, and A. Montanari. Universality in polytope phase transitions and message passing algorithms.Ann. Appl. Probab., 25(2):753–822, 2015. 25

2015
[9]

Bolthausen

E. Bolthausen. An iterative construction of solutions of the TAP equations for the Sherrington–Kirkpatrick model.Communications in Mathematical Physics, 325(1):333–366, 2014

2014
[10]

H.-B. Chen. On free energy of non-convex multi-species spin glasses.arXiv preprint arXiv:2411.13342, 2024

work page arXiv 2024
[11]

H.-B. Chen, V. Issa, and J.-C. Mourrat. The convex structure of the Parisi formula for multi-species spin glasses.arXiv preprint arXiv:2508.06397, 2025

work page arXiv 2025
[12]

Chen and S

W.-K. Chen and S. Tang. On convergence of the cavity and Bolthausen’s TAP iterations to the local magnetization.Comm. Math. Phys., 386(2):1209–1242, 2021

2021
[13]

Comets and J

F. Comets and J. Neveu. The Sherrington-Kirkpatrick model of spin glasses and stochastic calculus: the high temperature case.Comm. Math. Phys., 166(3):549–564, 1995

1995
[14]

Dey and Q

P.S. Dey and Q. Wu. Fluctuation results for multi-species Sherrington-Kirkpatrick model in the replica symmetric regime.J. Stat. Phys., 185(3):Paper No. 22, 40, 2021

2021
[15]

M. M. Dˇ zrbaˇ syan and A. B. Tavadyan. On weighted uniform approximation by polynomials of functions of several variables.Mat. Sb. (N.S.), 43(85):227–256, 1957

1957
[16]

O. Y. Feng, R. Venkataramanan, C. Rush, and R. J. Samworth. A unifying tutorial on Approximate Message Passing.Foundations and Trends in Machine Learning, 15(4):335– 536, 05 2022

2022
[17]

Guionnet, J

A. Guionnet, J. Ko, F. Krzakala, and L. Zdeborov´ a. Low-rank matrix estimation with inho- mogeneous noise.Information and Inference: A Journal of the IMA, 14(2), 04 2025

2025
[18]

W. Hachem. Approximate message passing for sparse matrices with application to the equi- libria of large ecological Lotka-Volterra systems.Stochastic Process. Appl., 170:Paper No. 104276, 34, 2024

2024
[19]

Jagannath, J

A. Jagannath, J. Ko, and S. Sen. Maxκ-cut and the inhomogeneous Potts spin glass.Ann. Appl. Probab., 28(3):1536–1572, 2018

2018
[20]

H. Kim. On the de Almeida–Thouless transition surface in the multi-species SK model with centered Gaussian external field.arXiv preprint arXiv:2509.18066, 2025

work page arXiv 2025
[21]

Liu and Z

Q. Liu and Z. Dong. Some rigorous results for the diluted multi-species SK model.J. Stat. Phys., 191(12):Paper No. 162, 25, 2024

2024
[22]

H. N. Mhaskar. Weighted polynomial approximation.J. Approx. Theory, 46(1):100–110,
[23]

Papers dedicated to the memory of G´ eza Freud
[24]

Panchenko.The Sherrington-Kirkpatrick model

D. Panchenko.The Sherrington-Kirkpatrick model. Springer Monographs in Mathematics. Springer, New York, 2013

2013
[25]

Panchenko

D. Panchenko. The free energy in a multi-species Sherrington-Kirkpatrick model.Ann. Probab., 43(6):3494–3513, 2015

2015
[26]

Talagrand.Mean field models for spin glasses

M. Talagrand.Mean field models for spin glasses. Volume I, volume 54 ofA Series of Modern Surveys in Mathematics. Springer, 2011. Basic examples

2011
[27]

Talagrand.Mean field models for spin glasses

M. Talagrand.Mean field models for spin glasses. Volume II, volume 55 ofA Series of Modern Surveys in Mathematics. Springer, 2011. Advanced replica-symmetry and low temperature

2011
[28]

Q. Wu. Thouless-Anderson-Palmer equations for the multi-species Sherrington-Kirkpatrick model.J. Stat. Phys., 191(7):Paper No. 87, 14, 2024. 26

2024