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arxiv: 2606.25925 · v1 · pith:BANSMOO5new · submitted 2026-06-24 · ❄️ cond-mat.dis-nn

Spectral properties and phase diagrams of sparse antagonistic random matrices with diagonal disorder and Jacobian-like structure

Luca Giammanco , Pietro Valigi , Chiara Cammarota This is my paper

Pith reviewed 2026-06-25 19:39 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn
keywords random matricesspectral propertiesphase diagramcavity methodpopulation dynamicsantagonistic interactionsdiagonal disorderJacobian structure
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The pith

Sparse antagonistic random matrices with diagonal disorder and Jacobian-like structure exhibit five distinct spectral phases.

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

The paper examines how diagonal disorder combined with a Jacobian-like structure modifies the spectra of sparse antagonistic random matrices. Without disorder, low connectivity produces a reentrance in the spectral boundary near the real axis that vanishes continuously with rising connectivity. Disorder introduces a segment of eigenvalues along the real axis that can cause the leading eigenvalue to jump discontinuously from complex to real. Using the cavity method and an adapted population dynamics algorithm, the authors construct a phase diagram containing five phases governed by the balance of connectivity and disorder. These spectral features control whether the underlying dynamical system remains stable, oscillates, or becomes unstable.

Core claim

In sparse antagonistic random matrices that also carry diagonal disorder and a Jacobian-like structure, the spectrum develops a segment of eigenvalues accumulating on the real axis. This segment can induce a discontinuous jump of the complex leading eigenvalue onto the real axis. The interplay of connectivity and disorder strength produces five distinct spectral phases, which are mapped by means of the cavity method together with an adaptation of the Population Dynamics algorithm.

What carries the argument

The cavity method combined with an adapted Population Dynamics algorithm that locates the boundary of the eigenvalue support and delineates the five phases.

If this is right

  • At low connectivity the reentrance effect near the real axis survives but is modified by the real-axis eigenvalue segment.
  • A continuous transition eliminates the reentrance once connectivity exceeds a threshold in the low-disorder regime.
  • A discontinuous transition appears once the real-axis segment reaches the leading eigenvalue.
  • The phase diagram is partitioned into five regions whose boundaries are set by the competition between connectivity and disorder strength.
  • The algorithm systematically underestimates the spectral support in the strong-disorder limit.

Where Pith is reading between the lines

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

  • The five phases may correspond to qualitatively different long-term behaviors in the dynamical systems these matrices describe, such as damped oscillations versus monotonic relaxation.
  • Direct large-scale diagonalization or moment methods could supply an independent check on the phase boundaries once the strong-disorder limitation is addressed.
  • Similar phase structures may appear in other sparse non-Hermitian ensembles once diagonal disorder and a Jacobian structure are imposed.

Load-bearing premise

The adapted population dynamics algorithm correctly identifies the spectral boundary even when diagonal disorder becomes strong.

What would settle it

Numerical diagonalization of finite matrices at strong disorder and intermediate connectivity, checking whether the actual eigenvalue support exceeds the boundary reported by the algorithm.

Figures

Figures reproduced from arXiv: 2606.25925 by Chiara Cammarota, Luca Giammanco, Pietro Valigi.

Figure 1
Figure 1. Figure 1: Examples of different shapes observed through direct diagonalisation of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Explicative diagram showing the differences between the matrix elements [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Results of the Population Dynamics algorithm (PD) and Direct Diagonali [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: An example of how our algorithm can fail for strong disorder in the [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Examples of all of the behaviours observed when changing the parame [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Phase diagrams of both matrix ensembles. The shaded areas highlight [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Examples of how we have located the emergence transition point for c [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Discontinuous transition at c=1.8 for Interaction-like matrices. The star￾shaped markers correspond to z∗, and we colour them red if the transition has not happened yet, while green otherwise. The vertical lines are placed at ℜ(z0 ). For both population sizes the results were averaged over Nm = 10 realisations. which the curves could be approximated as linear, to the larger ones, at which the curvature of … view at source ↗
Figure 9
Figure 9. Figure 9: (Re(z ∗ )−ℜ(z0 ))vs.d (or dx ) for the Interaction-like matrices at c = 1.8 and the Jacobian like ones at c = 2.1. The dash dotted lines represent the best fitting curves, while the full markers identify the transition points. E Locating the continuous transition Locating the continuous transition in d (dx ) through the study of the leading eigenvalue is somewhat challenging, as the border becomes flatter … view at source ↗
Figure 10
Figure 10. Figure 10: Continuous transition in the Jacobian-like case, for [PITH_FULL_IMAGE:figures/full_fig_p026_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Continuous transition for the Interaction-like matrices at [PITH_FULL_IMAGE:figures/full_fig_p027_11.png] view at source ↗
read the original abstract

Complex interacting systems are often modelled by random matrices whose spectral properties dictate stability. In sparse antagonistic matrices without diagonal disorder, low connectivity gives rise to a characteristic reentrance effect in the spectral boundary near the real axis, which disappears via a continuous transition as the connectivity increases. The reentrance effect implies the presence of a complex leading eigenvalue, which suggests the existence of a phase characterized by oscillatory dynamics around equilibrium. Here, we expand the investigation to matrices featuring diagonal disorder and a Jacobian-like structure. In these settings, the spectrum also develops a segment of eigenvalues accumulating on the real axis, which can trigger a discontinuous jump of the complex leading eigenvalue to a purely real value. The interplay between connectivity and disorder produces a rich variety of spectral behaviours. Employing the cavity method and a an adaptation of the Population Dynamics algorithm, we map a phase diagram with five distinct spectral phases. Finally, we show that the algorithm underestimates the spectral support under strong disorder, motivating future technical developments to handle this limit.

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 manuscript studies spectral properties of sparse antagonistic random matrices with diagonal disorder and Jacobian-like structure. Building on prior work showing reentrance in the spectral boundary for low connectivity without disorder, it employs the cavity method and an adapted population-dynamics algorithm to map a phase diagram in the connectivity-disorder plane that exhibits five distinct spectral phases, including regimes with complex leading eigenvalues and real-axis eigenvalue segments. The authors explicitly note that the algorithm underestimates spectral support under strong diagonal disorder.

Significance. If the reported phase boundaries prove robust, the work would provide a systematic characterization of how connectivity and disorder jointly control spectral features relevant to stability and oscillatory dynamics in complex systems. The application of standard cavity and population-dynamics techniques to this extended setting is a methodological strength, and the identification of multiple phases offers concrete, falsifiable predictions for the locations of transitions between spectral regimes.

major comments (2)
  1. [Abstract / results] Abstract and results section: The headline claim of five distinct spectral phases is delimited by the position of the complex leading eigenvalue and the real-axis accumulation segment. The manuscript states that the adapted population-dynamics algorithm underestimates the spectral support under strong diagonal disorder; because this bias directly shifts or erases the reported boundaries in the strong-disorder region of the phase diagram, the quantitative locations of at least two of the five phases remain unsupported without a corrected algorithm or explicit error bounds.
  2. [Methods] Methods: The adaptation of the population-dynamics algorithm for the Jacobian-like structure with diagonal disorder is presented as correctly locating the spectral boundary, yet the acknowledged underestimation in the strong-disorder limit indicates that the fixed-point equations or population sampling do not fully capture the support when disorder variance is large. A concrete test (e.g., comparison against exact diagonalization on small instances or an alternative solver) is needed to establish the regime of validity.
minor comments (1)
  1. [Model] Notation for the diagonal disorder distribution and the precise definition of the Jacobian-like structure should be stated explicitly in the model section to allow direct reproduction of the cavity equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the methodological contribution, and constructive critique. We agree that the acknowledged underestimation of spectral support by the population-dynamics algorithm in the strong-disorder regime limits the quantitative precision of phase boundaries there. We will revise the manuscript to clarify the regimes of validity, emphasize qualitative robustness of the five phases, and add explicit discussion of this limitation. Point-by-point responses to the major comments are below.

read point-by-point responses

  1. Referee: [Abstract / results] Abstract and results section: The headline claim of five distinct spectral phases is delimited by the position of the complex leading eigenvalue and the real-axis accumulation segment. The manuscript states that the adapted population-dynamics algorithm underestimates the spectral support under strong diagonal disorder; because this bias directly shifts or erases the reported boundaries in the strong-disorder region of the phase diagram, the quantitative locations of at least two of the five phases remain unsupported without a corrected algorithm or explicit error bounds.

    Authors: We acknowledge the underestimation bias in strong disorder, as already noted in the abstract and results. This implies that the precise numerical locations of boundaries involving the real-axis segment at high disorder are approximate rather than exact. However, the five phases are distinguished by qualitative changes in spectral structure (presence or absence of a complex leading eigenvalue and of a real-axis accumulation segment) that remain robustly observable. The transitions occur at moderate disorder values where the algorithm agrees well with the cavity equations. We will revise the abstract, results, and phase-diagram caption to state explicitly that quantitative boundary locations are reliable only for moderate disorder and that strong-disorder boundaries are indicative, thereby addressing the concern without requiring a new algorithm. revision: partial

  2. Referee: [Methods] Methods: The adaptation of the population-dynamics algorithm for the Jacobian-like structure with diagonal disorder is presented as correctly locating the spectral boundary, yet the acknowledged underestimation in the strong-disorder limit indicates that the fixed-point equations or population sampling do not fully capture the support when disorder variance is large. A concrete test (e.g., comparison against exact diagonalization on small instances or an alternative solver) is needed to establish the regime of validity.

    Authors: The cavity equations are exact in the thermodynamic limit; the population-dynamics implementation is a numerical solver whose sampling becomes incomplete for large disorder variance. We have performed limited comparisons with exact diagonalization on small finite systems in the moderate-disorder regime, finding agreement within finite-size effects. In the strong-disorder regime such comparisons are hindered by both the algorithm bias and pronounced finite-size corrections. We will add a new subsection or appendix reporting these validation tests together with a statement of the disorder range in which the algorithm is reliable, thereby establishing the regime of validity as requested. revision: yes

Circularity Check

0 steps flagged

No circularity; phase diagram obtained via standard independent methods

full rationale

The derivation applies the cavity method and an adapted Population Dynamics algorithm to compute spectral boundaries and the five-phase diagram directly from the random matrix ensemble. These techniques are standard in the literature and not defined in terms of the paper's own outputs or fitted parameters. No equations reduce a claimed prediction to a tautological fit, and no load-bearing step relies on self-citation chains or ansatzes smuggled from prior author work. The acknowledged underestimation of spectral support under strong disorder is a noted computational limitation affecting accuracy, not a circularity that makes the phase boundaries equivalent to their inputs by construction. The result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the cavity method and population-dynamics algorithm are treated as standard imported tools.

pith-pipeline@v0.9.1-grok · 5709 in / 1214 out tokens · 16159 ms · 2026-06-25T19:39:51.039133+00:00 · methodology

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