Localization and top eigenvalue detection

Benedikt Gr\"uger; Diego Tapias; Peter Sollich; Reimer K\"uhn

REVIEW 3 major objections 5 minor 27 references

Localization and top eigenvalue detection

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper establishes that for random matrices with a localized top eigenvector, the top eigenvalue is correctly located by the point where cavity precisions turn negative, not by the unit-growth-rate criterion of the standard method.

desk verdict A genuinely new and clean criterion for top eigenvalues with localized eigenvectors, backed by honest but narrow numerics and one unproved empirical assertion that needs referee attention. read the letter →

arxiv 2507.07310 v1 pith:RGOGBHHT submitted 2025-07-09 cond-mat.dis-nn cond-mat.stat-mech

Diego Tapias , Benedikt Grüger , Reimer Kühn , Peter Sollich This is my paper · ORCID

classification cond-mat.dis-nncond-mat.stat-mech

keywords topeigenvaluelocalizationcavitymethodAndersonmodelrandomregulargraphpopulationdynamicsspectraledgematrix

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper addresses the problem of locating the largest eigenvalue of a large sparse random matrix when the associated eigenvector is localized, as in the Anderson model. It shows that the standard cavity-method criterion—a unit growth rate of auxiliary fields—fails in this regime, either underestimating the eigenvalue (in population dynamics) or becoming very hard to detect numerically (in single instances). The authors propose a new criterion: the top eigenvalue is the smallest $\lambda$ for which all cavity precisions remain positive; below $\lambda_1$ at least one precision becomes negative. Using the Anderson model on random regular graphs, they show this estimator tracks the known top eigenvalue, with finite-size scaling similar to that of exact diagonalization.

What carries the argument

The central object is the set of cavity precisions $\Omega_j^{(i)}(\lambda)$, the inverse variances of the Gaussian ansatz for the cavity distributions, determined by the self-consistent equation $\Omega_j^{(i)} = \lambda - J_{jj} - \sum_{l \in \partial j \setminus i} J_{jl}^2 / \Omega_l^{(j)}$ (eq. 6). The new criterion (eq. 12) uses the constraint that all precisions must be positive for the cavity measure to be well defined: $\lambda_u$ is the smallest $\lambda$ such that every precision is positive. The mechanism is that, for a localized top eigenvector, decreasing $\lambda$ below $\lambda_1$ forces at least one precision through zero to a negative value, so the positivity threshold marks $\lambda_1$. This replaces the growth-rate criterion based on the non-backtracking operator (eq. 8) and the auxiliary fields $H$, which the paper shows remains satisfied at unphysical negative precisions.

What would settle it

Exhibit a random matrix ensemble with a provably localized top eigenvector for which the cavity precisions do not all stay positive above the true $\lambda_1$, or for which negative precisions first appear at a $\lambda$ strictly above $\lambda_1$; then $\lambda_u$ would be biased. A concrete test would be a model with a localized top eigenvector that is an outlier separated from the bulk by a spectral gap, a case the paper does not consider, checking whether the positivity threshold still coincides with $\lambda_1$.

Watch

Extended reading notes

Core claim

The central claim is that for a random matrix whose top eigenvector is localized, the estimator $\lambda_u = \min\{\lambda \in \mathbb{R} : \Omega_j^{(i)}(\lambda) > 0 \, \forall \text{ directed edges}\}$ gives a reliable estimate of the top eigenvalue $\lambda_1$, while the growth-rate criterion $\eta = 1$ does not. The paper states the basis of this as the finding that solving equation (6) for $\lambda < \lambda_1$ yields at least one negative cavity precision whenever the top eigenvector is localized. The growth-rate criterion is shown to remain satisfied at $\lambda$ values where the cavity precisions are already negative and therefore unphysical, so it is meaningless in the localized regime; the new criterion instead uses the appearance of negative precisions as the marker of $\lambda_1$.

Load-bearing premise

The load-bearing assumption, stated as a numerical finding rather than a derivation, is that for a localized top eigenvector every cavity precision is positive for all $\lambda$ above the true top eigenvalue and at least one becomes negative as soon as $\lambda$ drops below it, so that the threshold in eq. (12) exactly marks $\lambda_1$.

Editorial extensions

If this is right

For ensembles whose top eigenvector is localized, the growth-rate criterion should not be used: it can remain satisfied at $\lambda$ values where the cavity precisions are unphysically negative.
The new positivity criterion locates $\lambda_1$ for single instances as well as for population-dynamics solutions of the thermodynamic limit.
For the Anderson model on random regular graphs, the estimator $\lambda_u$ scales with population size in the same way that the exact diagonalization result scales with system size, confirming its reliability.
The method applies to any random matrix ensemble with mobility edges, since those have a localized top eigenvector by the argument given in the introduction.
Reliable estimation of the top eigenvalue enables the detection of dynamical free energies in biased stochastic systems, as noted in the outlook.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The criterion might be interpretable as a stability or spinodal condition of the Gaussian cavity ansatz: negative precisions signal the breakdown of the Gaussian approximation, so $\lambda_u$ would be the point where the cavity measure first becomes ill-defined. The paper does not develop this interpretation.
The paper only tests the criterion when the localized top eigenvector sits at the spectral edge (the Anderson model). A natural extension is to test ensembles with a localized outlier separated from the bulk, or with a mobility edge inside the spectrum, to see whether the positivity threshold still coincides with $\lambda_1$.
Because the population-size scaling mirrors the instance-size scaling, the estimator could be used to extrapolate the thermodynamic-limit top eigenvalue and, potentially, to locate mobility edges, but the paper notes that the finite-size corrections limit the advantage over exact diagonalization.
The paper leaves open the distribution of top eigenvector components for localized states; a future method that also captures this distribution could build on the positivity criterion.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. The paper isolates a genuine failure mode of the growth-rate criterion for top eigenvalues with localized eigenvectors, and proposes a simple alternative: estimate λ1 as the smallest λ for which all cavity precisions stay positive. The idea is clean and, as far as the paper's own discussion shows, new. The numerics are honest and the empirical scaling is suggestive. But the central criterion is exactly that—an empirical finding, not a derived statement—and the validation on one model family (Anderson model on RRG, W=12, c=3) leaves real room for a model-specific coincidence.

What's good. The demonstration that the growth-rate criterion fails in the thermodynamic-limit population dynamics because the narrow peak at λ1 is washed out, while it still works for single instances, is a useful diagnostic. The contrast between Fig. 2 and Fig. 3 makes the mechanism clear. The finite-size scaling analysis is careful: they fit c0 + c1 N^-α1, give honest error bars, and explicitly note that population-size effects are of the same order as instance-size effects. The paper also openly flags the unresolved problem of eigenvector component statistics for localized top eigenvectors. That is the right kind of self-assessment.

Where it is soft. Eq. (12) rests on the assertion that for a localized top eigenvector, precisions are positive for λ ≥ λ1 and at least one goes negative just below λ1. The paper calls this a finding and gives no derivation. The connection to the real-energy analysis of Ref. [20], which studies the same cavity precisions via G=1/Ω, is not made; the mechanism is therefore not independently supported. The numerical validation has genuine slack: the extrapolated thermodynamic-limit value is 8.6 ± 0.3 against the theoretical 8.83, so the agreement is within error but not sharp. And because in the Anderson model the top eigenvalue coincides with the spectral edge, the criterion might be detecting the edge where precisions lose positivity rather than λ1 as such. A second localized model with a gap, or an analytical argument, would settle this. None of these are fatal, but they are real.

Who this is for. People working on cavity-method estimates of extreme eigenvalues, large deviations in stochastic dynamics, and Anderson localization on trees. The paper deserves a serious referee: the idea is simple, useful, and testable, and the weaknesses are the kind a revision can address. I would send it out and ask for either a proof sketch of the positivity threshold or a demonstration on a second model with a separated localized top eigenvalue.

Referee Report

3 major / 5 minor

Summary. The paper addresses the problem of detecting the top eigenvalue and its associated eigenvector for sparse random matrices when the top eigenvector is localized. The existing real-valued cavity method uses a growth-rate criterion, η=1 in eq. (11), to locate λ1. The authors show numerically that this criterion underestimates λ1 for the Anderson model on random regular graphs, because in population dynamics the growth-rate peak is washed out. They propose a new criterion, λu in eq. (12), defined as the smallest λ for which all cavity precisions are positive, and present numerical evidence from population dynamics and finite single instances that λu approaches the theoretical λ1 ≈ 8.83 for W=12, c=3. They also show scaling of the estimator with system size and population size, and discuss the relation to Lifshitz tails and the spectral edge.

Significance. If the new criterion is correct, it fills a gap in the cavity-method toolbox for top eigenpair detection and could be applied to large-deviation problems and biased stochastic dynamics where localized top eigenvectors appear. The paper provides a useful comparison between single-instance and population-dynamics behavior, and the scaling analysis in Fig. 4 is a constructive step toward validation. These are genuine strengths. However, the central assertion behind eq. (12) is presented as an empirical finding rather than a derived result, and the quantitative validation is not sharp because it relies on a single model and a wide error bar.

major comments (3)

[Real-valued cavity method, eq. (12)] The estimator λu rests entirely on the assertion, stated after eq. (11), that 'solving equation (6) for λ < λ1 yields at least one negative cavity precision whenever the top eigenvector is localized.' This is presented as a 'finding' without proof or reference. Since the entire paper depends on this property, it is load-bearing and must be either proved or clearly labelled as a conjecture supported by numerical evidence on a single model family. The paper should also connect to Ref. [20], which analyses the same cavity precisions via propagators G=1/Ω with real energies; if the positivity threshold there is known to be the spectral edge, that connection would provide independent support for the mechanism.
[Finite size (and population size) effects, Fig. 4] The quantitative validation of λu→λ1 is based on a three-parameter fit λ1 = c0 + c1 N^{-α1} that yields c0 = 8.6 ± 0.3, whose one-sigma interval only marginally covers the theoretical value 8.83. The paper does not report the corresponding extrapolated value for the population-dynamics estimator λu(NP) or its uncertainty; the statement that the two scalings are 'essentially the same' is qualitative. Please provide explicit extrapolated values with error bars for both exact diagonalization and population dynamics, and test the robustness of the fit by varying the fit range or including higher-order corrections.
[Population dynamics, Figs. 1–2] For the Anderson model on a random regular graph there is no spectral gap, so the top eigenvalue coincides with the spectral edge. The proposed criterion could therefore be detecting the spectral edge (where precision positivity is lost) rather than λ1 specifically. To establish that eq. (12) is a criterion for the top eigenvalue of a localized top eigenvector, rather than merely an edge detector, the authors should test a model in which the spectral edge and λ1 are distinct, or provide an analytical argument that the positivity threshold equals λ1 in the localized regime. Without such a test, the agreement in Fig. 2 may be a model-specific coincidence.

minor comments (5)

[Captions of Figs. 1, 2] The labels 'NP = 105' should read 'NP = 10^5'; similarly, 'δ = 10 −1' in Fig. 1 should be '10^{-1}'.
[Introduction, second paragraph] The phrase 'i.e. values that separate extended from localized eigenvectors in the bulk of the spectrum [15–17]):' has an unmatched parenthesis; please fix the punctuation.
[Eq. (12) and notation] The set of edges is denoted E, which can be confused with on-site energies Ei; consider using a different symbol for edges.
[Appendix, eq. (16)] The expression '1/Ω(j) l' is notationally unclear; it should be written as 1/Ω_l^{(j)} to match the superscript conventions used in eq. (6).
[References] Reference [24] is a bare URL to the Arpack.jl library; please provide a proper citation with author and version information.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proposed estimator is defined directly from cavity precisions and is validated against independent theoretical and exact-diagonalization benchmarks.

full rationale

The paper's new criterion λu = min{λ : Ω_i(λ) > 0 for all directed edges} (eq. 12) is not fitted to λ1. It is computed from the self-consistent cavity equations (6), and the paper's claim that λu estimates λ1 rests on the stated empirical 'finding' that negative precisions appear only below λ1 for localized top eigenvectors. This is an input assumption, not a parameter fitted to the target eigenvalue, and the paper does not define λu in terms of λ1. Validation is performed against the independent theoretical result λ1 = 2√(c−1) + W/2 (eq. 15) from Refs. [26,27] and against exact diagonalization via Lanczos for finite instances. The growth-rate criterion is used as a baseline, and its proof is cited to Ref. [9], which has no overlapping authors with the present paper. The self-citations (Refs. [10], [18], [12]) describe the real-valued cavity method and its derivation; the relevant cavity equations are re-derived in the present paper, and the new criterion does not reduce to those citations. The main limitation is that the central 'finding' is demonstrated numerically only for the Anderson model on random regular graphs, and the thermodynamic-limit scaling estimate has a one-sigma range that just covers the theoretical value (8.6 ± 0.3 vs 8.83). This is a correctness risk or an incompleteness in the proof, not a circular derivation. No equation or estimator in the paper reduces by construction to its own inputs, and no load-bearing argument relies on a self-citation for the new result. Hence the appropriate circularity score is 0.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central estimator λ_u has no fitted parameters; it is computed directly from the cavity precisions. The only fitted quantities are the parameters of the finite-size scaling curve used for validation. The main axioms are the standard cavity-method approximations and the externally derived benchmark value for the Anderson model.

free parameters (3)

c0 = 8.6 ± 0.3
Extrapolated thermodynamic-limit top eigenvalue from the fit λ1 = c0 + c1 N^{-α1} in Fig. 4; used as the numerical benchmark for validating the new criterion.
c1 = not reported
Amplitude in the same finite-size scaling fit in Fig. 4.
α1 = not reported
Exponent in the same finite-size scaling fit in Fig. 4.

assumptions (4)

domain assumption Gaussian ansatz for cavity distributions (eq. 5)
The cavity method assumes Gaussian cavity distributions; this is standard for the real-valued cavity method but is an approximation for finite connectivity.
domain assumption Local tree-likeness of random regular graphs
The cavity equations assume the graph is locally tree-like; random regular graphs satisfy this in the limit of large size.
standard math Theoretical top eigenvalue λ1 = 2√(c-1) + W/2 (eq. 15)
Taken from Refs. [14,26,27] as an external benchmark for the Anderson model on the Bethe lattice; the paper does not rederive it.
domain assumption Inverted thermodynamic limit equivalence
The real-valued cavity method is claimed equivalent to the 'inverted thermodynamic limit' of Refs. [19,20]; this underpins the interpretation of population dynamics results.

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Cite this review

Pith. "Pith review of Localization and top eigenvalue detection." pith.science (2026). https://pith.science/paper/RGOGBHHT

@misc{pith2026250707310,
  author       = {Pith},
  title        = {Pith review of: Localization and top eigenvalue detection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RGOGBHHT}},
  note         = {Machine review of arXiv:2507.07310}
}

read the original abstract

The detection of the top eigenvalue and its corresponding eigenvector in ensembles of random matrices has significant applications across various fields. An existing method, based on the linear stability of a complementary set of cavity equations, has been successful in identifying the top eigenvalue when the associated eigenvector is extended. However, this approach fails when the eigenvector is localized. In this work, we adapt the real-valued cavity method to address this limitation by introducing a novel criterion that exploits the constraints of the cavity equations to detect the top eigenvalue in systems with a localized top eigenvector. Our results are validated using the Anderson model as a paradigmatic example.

Figures

Figures reproduced from arXiv: 2507.07310 by the authors.

**Figure 4.** Dependence of the top eigenvalue with system [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [20]

Anderson tran- sition on the bethe lattice: an approach with real ener- gies

Giorgio Parisi, Saverio Pascazio, Francesca Pietracaprina, Valentina Ros, and Antonello Scardicchio. Anderson tran- sition on the bethe lattice: an approach with real ener- gies. Journal of Physics A: Mathematical and Theoretical, 53(1):014003, 2019

work page 2019

[2] [1]

Exact large devia- tion function in the asymmetric exclusion process

Bernard Derrida and Joel L Lebowitz. Exact large devia- tion function in the asymmetric exclusion process. Phys- ical review letters, 80(2):209, 1998

work page 1998

[3] [2]

A spectral heuristic for bisecting ran- dom graphs

Amin Coja-Oghlan. A spectral heuristic for bisecting ran- dom graphs. Random Structures & Algorithms, 29(3):351– 398, 2006

work page 2006

[4] [3]

Google’s PageR- ank and beyond: The science of search engine rankings

Amy N Langville and Carl D Meyer. Google’s PageR- ank and beyond: The science of search engine rankings. Princeton university press, 2006

work page 2006

[5] [4]

Weighted percolation on directed networks

Juan G Restrepo, Edward Ott, and Brian R Hunt. Weighted percolation on directed networks. Physical re- view letters, 100(5):058701, 2008

work page 2008

[6] [5]

Estimating the principal components of correlation matrices from all their empirical eigenvectors

R´ emi Monasson and Dario Villamaina. Estimating the principal components of correlation matrices from all their empirical eigenvectors. Europhysics Letters, 112(5):50001, 2015

work page 2015

[7] [6]

May’s instabil- ity in large economies

Jos´ e Moran and Jean-Philippe Bouchaud. May’s instabil- ity in large economies. Physical Review E, 100(3):032307, 2019

work page 2019

[8] [7]

Ergodicity and large deviations in physical systems with stochastic dynamics

Robert L Jack. Ergodicity and large deviations in physical systems with stochastic dynamics. The European Physical Journal B, 93:1–22, 2020

work page 2020