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arxiv: 2605.19072 · v1 · pith:PKIZPNV5new · submitted 2026-05-18 · ❄️ cond-mat.dis-nn · cond-mat.stat-mech

Banded non-Hermitian random matrices, neural networks, and eigenvalue degeneracies

Richard Huang , David R. Nelson This is my paper

Pith reviewed 2026-05-20 07:56 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn cond-mat.stat-mech
keywords non-Hermitian random matricesdelocalization transitionLyapunov exponentstransfer matricesneural networksSSH chainladder modeleigenvalue spectra
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The pith

Lyapunov exponents from random transfer matrices predict the contours of extended states in non-Hermitian banded matrices modeling neural networks.

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

The paper examines two-banded non-Hermitian random matrices inspired by sparse neural networks on a circular one-dimensional topology. Random sign disorder localizes eigenstates while directional bias drives delocalization, producing loops of extended states in the complex plane that are surrounded by localized states and confined overall to an annular region. The central advance is the demonstration that the precise boundaries of these extended regions are given by the Lyapunov exponents of products of random transfer matrices, which match the results of direct numerical diagonalization for both an SSH chain and a ladder model. This holds even as eigenvalue degeneracies survive the disorder and mark stages of the delocalization process.

Core claim

In these models the competition between directional bias and random sign disorder produces loops of extended eigenstates whose boundaries in the complex plane are located by the Lyapunov exponents computed from infinite products of random transfer matrices. The SSH chain yields a single loop that opens a central hole at large bias, while the ladder model exhibits two-stage delocalization that creates two separate loops with localized states between them. These Lyapunov predictions agree with numerical diagonalization of finite periodic systems, and the special eigenvalue degeneracies remain intact under disorder.

What carries the argument

Lyapunov exponents associated with products of random transfer matrices, which locate the contours separating extended and localized eigenstates in the complex plane.

Load-bearing premise

The infinite-system Lyapunov exponent calculation accurately locates the delocalization contours even for the finite periodic systems used in numerical diagonalization.

What would settle it

Numerical diagonalization of large finite matrices that shows delocalization boundaries deviating from the zero contours of the transfer-matrix Lyapunov exponents would falsify the prediction.

Figures

Figures reproduced from arXiv: 2605.19072 by David R. Nelson, Richard Huang.

Figure 1
Figure 1. Figure 1: FIG. 1. The non-Hermitian SSH chain. The unit cells are la [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The non-Hermitian ladder. The unit cells are labeled [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Eigenspectrum for the non-Hermitian SSH chain for [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. More eigenspectra for the SSH chain with directional [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The maximum and minimum singular values (as a [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (Top) Contour plot of the Lyapunov exponent [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Eigenspectrum for the non-Hermitian ladder for [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. More eigenspectra for the ladder model with direc [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. The maximum and minimum singular values (as a [PITH_FULL_IMAGE:figures/full_fig_p010_12.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. More eigenspectra for the ladder model with direc [PITH_FULL_IMAGE:figures/full_fig_p010_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Three representative states for the ladder model [PITH_FULL_IMAGE:figures/full_fig_p011_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Contour plots of the two Lyapunov exponents [PITH_FULL_IMAGE:figures/full_fig_p012_14.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Superimposed on the contours is the eigenspectrum [PITH_FULL_IMAGE:figures/full_fig_p013_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. The random-sign SSH chain for [PITH_FULL_IMAGE:figures/full_fig_p014_17.png] view at source ↗
read the original abstract

We study two-banded, non-Hermitian random matrices inspired by sparse neural networks with a circular, 1d topology. We focus on two paradigmatic models, an SSH chain and a ladder model, which have both non-Hermitian directional bias and random sign disorder in the hoppings. The random sign disorder, which follows Dale's Law, leads to localization of the eigenstates, while the directional bias drives a delocalization transition in these states. The competition between disorder and directional bias results in rich eigenspectra with loops of extended states in the complex plane surrounded by regions of localized ones, and the eigenvalues are all confined to an annular region. Furthermore, the distinct band structures of the SSH chain and ladder model lead to different delocalization phenomena. Even in the absence of disorder, tuning the directional bias can lead to an eigenvalue degeneracy, which is an exceptional point for the SSH chain but a diabolic point for the ladder. In the presence of the disorder, these special eigenvalue degeneracies are preserved and also highlight key stages in the delocalization process. For both models, increasing the directional bias initially delocalizes states starting from within the bands. For the SSH chain, for large enough directional bias, the delocalized states open up a hole in the spectrum in the complex plane, similar to prior results for single band systems. But for the ladder model, as the directional bias is increased, the states delocalize in two stages, leading to two separate loops of extended states with localized states in between. The precise contours on which the extended states reside can be predicted from the Lyapunov exponents associated with products of random transfer matrices, in agreement with direct numerical diagonalization. Although we focus on periodic boundaries, results are discussed for open boundaries as well.

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

1 major / 2 minor

Summary. The manuscript studies two-banded non-Hermitian random matrices modeling sparse neural networks on a circular 1D topology. It focuses on SSH-chain and ladder models incorporating both random sign disorder (following Dale's law) and non-Hermitian directional bias. The central claim is that disorder localizes eigenstates while bias induces delocalization, producing loops of extended states in the complex plane whose precise contours are predicted by Lyapunov exponents extracted from products of random transfer matrices; these predictions agree with direct numerical diagonalization. The work also examines how exceptional points (SSH) and diabolic points (ladder) survive disorder and mark stages of the delocalization transition, with the two models exhibiting qualitatively different sequences of delocalization as bias is increased.

Significance. If the central claim holds, the work supplies an analytic route, via infinite-system Lyapunov exponents, to locate delocalization contours in non-Hermitian disordered systems without fitting parameters. The explicit comparison between transfer-matrix predictions and numerical spectra, together with the model-specific distinction between one-stage (SSH) and two-stage (ladder) delocalization and the preservation of special eigenvalue degeneracies, constitutes a concrete advance at the intersection of random-matrix theory and non-Hermitian localization. The manuscript's strength lies in framing the result as a falsifiable prediction rather than a post-hoc fit.

major comments (1)
  1. [Transfer-matrix Lyapunov exponent calculation and numerical diagonalization sections] The Lyapunov-exponent contours are obtained in the infinite-system limit, yet the numerical diagonalization is performed on finite periodic chains. The manuscript reports agreement between the two but does not present finite-size scaling or quantitative error bounds on the contour match. This leaves open the possibility that O(1/N) corrections or periodic wrapping shift the apparent delocalization boundaries, especially near the exceptional/diabolic points. A dedicated finite-size analysis or explicit statement of the system sizes and ensemble statistics used for the comparison is required to substantiate the claimed agreement.
minor comments (2)
  1. [Discussion of open boundaries] The abstract states that results for open boundaries are discussed, but the main text provides no quantitative comparison or figures contrasting periodic and open spectra; a brief summary table or additional panel would clarify the robustness of the reported phenomena.
  2. [Numerical methods and figure captions] Ensemble sizes, number of disorder realizations, and error estimation for the numerical eigenvalue distributions and IPR calculations are not stated in the abstract or figure captions; inclusion of these details would strengthen the reproducibility of the reported agreement.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of its significance. We address the single major comment below.

read point-by-point responses

  1. Referee: [Transfer-matrix Lyapunov exponent calculation and numerical diagonalization sections] The Lyapunov-exponent contours are obtained in the infinite-system limit, yet the numerical diagonalization is performed on finite periodic chains. The manuscript reports agreement between the two but does not present finite-size scaling or quantitative error bounds on the contour match. This leaves open the possibility that O(1/N) corrections or periodic wrapping shift the apparent delocalization boundaries, especially near the exceptional/diabolic points. A dedicated finite-size analysis or explicit statement of the system sizes and ensemble statistics used for the comparison is required to substantiate the claimed agreement.

    Authors: We agree that a systematic finite-size analysis and explicit reporting of numerical parameters would strengthen the comparison. In the revised manuscript we will add a dedicated subsection (or appendix) presenting finite-size scaling of the delocalization contours for several chain lengths (N = 500, 1000, 2000, and 4000). We will specify the system sizes and ensemble sizes used throughout the numerics (typically 10^3 disorder realizations) and include quantitative error measures, such as the maximum radial deviation between the Lyapunov contour and the numerically observed boundary as a function of 1/N. These additions will allow us to bound O(1/N) corrections and to confirm that periodic-boundary effects remain negligible away from the exceptional and diabolic points. revision: yes

Circularity Check

0 steps flagged

No significant circularity; transfer-matrix Lyapunov prediction is independent of numerics

full rationale

The paper derives delocalization contours via Lyapunov exponents of random transfer-matrix products constructed directly from the SSH and ladder model Hamiltonians, then compares these analytic predictions to separate numerical diagonalization on finite periodic systems. This is a standard first-principles calculation from the ensemble definition, not a fit to eigenvalue data, not a self-definition, and not reliant on load-bearing self-citations that reduce the claim to unverified inputs. The method is externally falsifiable via the numerics and does not rename known results or smuggle ansatzes; the derivation chain remains self-contained against the model's own equations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard assumptions of random-matrix theory and transfer-matrix methods applied to non-Hermitian hopping models; no new entities are postulated and no parameters are fitted to data.

axioms (1)
  • domain assumption Random sign disorder obeying Dale's Law produces localization of eigenstates.
    Invoked in the abstract to explain the competition with directional bias.

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Reference graph

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