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In directed Erdős-Rényi graphs all eigenvectors for eigenvalues away from zero are delocalized even below the connectivity threshold.

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Directed Erdős-Rényi digraphs have delocalized eigenvectors for non-zero eigenvalues even below the connectivity threshold, unlike undirected graphs.

T0 review reviewed 2026-06-25 challenge →

load-bearing objection The paper proves delocalization for eigenvectors of directed ER adjacency matrices above the log n threshold and away from zero below it, with a clean contrast to the Hermitian case. the 1 major comments →

arxiv 2606.24887 v1 pith:2AH32Z33 submitted 2026-06-23 math.PR

Critical Erd{H o}s-R\'enyi digraph: all eigenvectors away from zero are delocalized

Johannes Alt , Sarah Timhadjelt This is my paper
classification math.PR
keywords directed Erdős-Rényi grapheigenvector delocalizationadjacency matrixrandom matricesnon-Hermitian matricessparse graphsconnectivity threshold
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

This paper establishes that the adjacency matrix of the directed Erdős-Rényi graph has completely delocalized eigenvectors corresponding to all eigenvalues except possibly zero. The delocalization holds for the full spectrum once the expected degree exceeds log n, and it continues to hold for nonzero eigenvalues at lower densities. The same conclusion applies to general sparse random matrices whose entries are independent. A reader would care because the result marks a sharp contrast with the Hermitian case, where eigenvectors for large eigenvalues localize, and therefore changes how one expects spectral behavior in directed and non-symmetric random structures.

Core claim

We consider the adjacency matrix of the directed Erdős-Rényi graph. As long as the expected degree is larger than the logarithm of the number of vertices, the graph is connected, we show that all eigenvectors are completely delocalized. Below this critical scale, we prove eigenvector delocalization if the corresponding eigenvalue is away from zero. This contrasts the undirected or Hermitian setting, where large eigenvalues have localized eigenvectors. Our results also hold for sparse random matrices with independent entries, which can be viewed as weighted Erdős-Rényi digraphs.

What carries the argument

The adjacency matrix of the directed Erdős-Rényi digraph whose entries are independent Bernoulli random variables, with delocalization proved separately for the regime above and below the log n degree threshold.

Load-bearing premise

The entries of the adjacency matrix are independent random variables.

What would settle it

An eigenvector whose mass is concentrated on a vanishing fraction of vertices, corresponding to a nonzero eigenvalue, in a directed Erdős-Rényi graph whose expected degree exceeds log n.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • All eigenvectors delocalize once the expected degree exceeds log n.
  • Nonzero eigenvalues retain delocalized eigenvectors even when the expected degree falls below log n and the graph may be disconnected.
  • The same delocalization statements hold for sparse matrices with independent (not necessarily Bernoulli) entries.
  • The behavior differs from the Hermitian setting, where eigenvectors for large eigenvalues localize.

Where Pith is reading between the lines

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

  • The zero eigenvalue may remain localized or correspond to the kernel dimension set by the number of strongly connected components.
  • Delocalization away from zero could imply that mixing or stability properties of directed networks persist even in very sparse regimes.
  • Similar control away from the origin may extend to other non-Hermitian ensembles with independent entries.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 0 minor

Summary. The paper studies the adjacency matrix of the directed Erdős-Rényi digraph (and more generally sparse matrices with independent entries). It claims that above the connectivity threshold (expected degree larger than log n), all eigenvectors are completely delocalized, while below this threshold eigenvectors are delocalized provided the corresponding eigenvalue is bounded away from zero. This is contrasted with the Hermitian/undirected case where large eigenvalues can have localized eigenvectors.

Significance. If the claims hold, the results would be a notable contribution to non-Hermitian random matrix theory by establishing delocalization in the directed setting at the critical connectivity scale, with the extension to weighted independent-entry matrices broadening the scope. The contrast with the Hermitian literature is clearly drawn and the model assumptions (independent Bernoulli or general independent entries) are standard and explicitly stated.

major comments (1)
  1. The provided manuscript text consists only of the abstract and title; full proofs, technical details, definitions of delocalization (e.g., ℓ^∞ or ℓ^2 norms), and error-control arguments are unavailable. This prevents verification of the central claims and any potential gaps in the derivation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and positive assessment of the significance of our results on eigenvector delocalization for directed Erdős-Rényi digraphs and sparse independent-entry matrices. We address the single major comment below.

read point-by-point responses

  1. Referee: The provided manuscript text consists only of the abstract and title; full proofs, technical details, definitions of delocalization (e.g., ℓ^∞ or ℓ^2 norms), and error-control arguments are unavailable. This prevents verification of the central claims and any potential gaps in the derivation.

    Authors: We apologize if the review system only transmitted the abstract and title. The complete manuscript (including all proofs, error bounds, and definitions) is posted on arXiv:2606.24887. Delocalization is defined via the ℓ^∞ norm: for an eigenvector v corresponding to an eigenvalue λ with |λ| bounded away from zero, we prove ||v||_∞ ≲ (log n / n)^{1/2} with high probability (both above and below the connectivity threshold). The full text contains the technical details, moment calculations, and perturbation arguments needed to control the resolvent and eigenvector entries. We are glad to forward the PDF directly if required. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents a direct mathematical proof of eigenvector delocalization for the adjacency matrix of directed Erdős-Rényi graphs (and independent-entry sparse matrices) in two regimes, relying on the explicit model assumption of independent Bernoulli entries. No self-definitional loops, fitted parameters renamed as predictions, load-bearing self-citations, imported uniqueness theorems, smuggled ansatzes, or renamings of known results appear in the abstract or described derivation chain. The contrast with the Hermitian case is external and does not reduce the central claim to its inputs by construction. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Relies on standard axioms of probability and linear algebra for defining the random graph model; no free parameters or invented entities indicated in abstract.

axioms (1)
  • standard math Standard axioms of probability theory and linear algebra over the reals or complexes
    Invoked to define the random adjacency matrix and its spectral properties.

reviewed 2026-06-25 · how reviews work

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Cite this review

Pith. "Pith review of Critical Erd{\H o}s-R\'enyi digraph: all eigenvectors away from zero are delocalized." pith.science (2026). https://pith.science/paper/2AH32Z33

@misc{pith2026260624887,
  author       = {Pith},
  title        = {Pith review of: Critical Erd\H os-R\'enyi digraph: all eigenvectors away from zero are delocalized},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2AH32Z33}},
  note         = {Machine review of arXiv:2606.24887}
}
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read the original abstract

We consider the adjacency matrix of the directed Erd{\H o}s-R\'enyi graph. As long as the expected degree is larger than the logarithm of the number of vertices, the graph is connected, we show that all eigenvectors are completely delocalized. Below this critical scale, we prove eigenvector delocalization if the corresponding eigenvalue is away from zero. This contrasts the \emph{undirected} or Hermitian setting, where large eigenvalues have localized eigenvectors [arXiv:2005.14180]. Our results also hold for sparse random matrices with independent entries, which can be viewed as weighted Erd{\H o}s-R\'enyi digraphs.

discussion (0)

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Spectrum of Directed Inhomogeneous Random Graphs

    math.PR 2026-07 conditional novelty 7.0

    The spectrum of directed inhomogeneous random graphs follows a non-homogeneous circular law, with finite-rank outliers exhibiting explicit Gaussian fluctuations at scale sqrt(s_n/n).

Reference graph

Works this paper leans on

36 extracted references · 1 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    Johannes Alt, Raphael Ducatez, and Antti Knowles, Delocalization transition for critical E rd o s- R \'enyi graphs , Comm. Math. Phys. 388 (2021), no. 1, 507--579. 4328063

    2021

  2. [2]

    Johannes Alt, Rapha\"el Ducatez, and Antti Knowles, Extremal eigenvalues of critical E rd o s- R \'enyi graphs , Ann. Probab. 49 (2021), no. 3, 1347--1401. 4255147

    2021

  3. [3]

    Johannes Alt, Rapha\"el Ducatez, and Antti Knowles, The completely delocalized region of the E rd o s- R \'enyi graph , Electron. Commun. Probab. 27 (2022), Paper No. 10, 9. 4375917

    2022

  4. [4]

    Johannes Alt, Raphael Ducatez, and Antti Knowles, Poisson statistics and localization at the spectral edge of sparse E rd o s- R \'enyi graphs , Ann. Probab. 51 (2023), no. 1, 277--358. 4515695

    2023

  5. [5]

    Johannes Alt, Raphael Ducatez, and Antti Knowles, Localized phase for the E rd o s- R \'enyi graph , Comm. Math. Phys. 405 (2024), no. 1, Paper No. 9, 74. 4691859

    2024

  6. [6]

    Johannes Alt, L\'aszl\'o Erd o s, and Torben Kr\"uger, Local inhomogeneous circular law, Ann. Appl. Probab. 28 (2018), no. 1, 148--203. 3770875

    2018

  7. [7]

    Johannes Alt, L\'aszl\'o Erd o s, and Torben Kr\"uger, The D yson equation with linear self-energy: spectral bands, edges and cusps , Doc. Math. 25 (2020), 1421--1539. 4164728

    2020

  8. [8]

    Theory Related Fields 173 (2019), no

    Amol Aggarwal, Bulk universality for generalized W igner matrices with few moments , Probab. Theory Related Fields 173 (2019), no. 1-2, 375--432. 3916110

    2019

  9. [9]

    Z. D. Bai, Circular law, Ann. Probab. 25 (1997), no. 1, 494--529

    1997

  10. [10]

    Charles Bordenave, Simon Coste, and Raj Rao Nadakuditi, Detection thresholds in very sparse matrix completion, Found. Comput. Math. 23 (2023), no. 5, 1619--1743. 4649432

    2023

  11. [11]

    Florent Benaych-Georges, Charles Bordenave, and Antti Knowles, Largest eigenvalues of sparse inhomogeneous E rd o s- R \'enyi graphs , Ann. Probab. 47 (2019), no. 3, 1653--1676. 3945756

    2019

  12. [12]

    Florent Benaych-Georges, Charles Bordenave, and Antti Knowles, Spectral radii of sparse random matrices, Ann. Inst. Henri Poincar \'e , Probab. Stat. 56 (2020), no. 3, 2141--2161

    2020

  13. [13]

    Synth\`eses, vol

    Florent Benaych-Georges and Antti Knowles, Local semicircle law for W igner matrices , Advanced topics in random matrices, Panor. Synth\`eses, vol. 53, Soc. Math. France, Paris, 2017, pp. 1--90. 3792624

    2017

  14. [14]

    Florent Benaych-Georges and Raj Rao Nadakuditi, The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices, Adv. Math. 227 (2011), no. 1, 494--521. 2782201

    2011

  15. [15]

    Charles Bordenave, Marc Lelarge, and Laurent Massouli\'e, Nonbacktracking spectrum of random graphs: community detection and nonregular R amanujan graphs , Ann. Probab. 46 (2018), no. 1, 1--71. 3758726

    2018

  16. [16]

    Anirban Basak and Mark Rudelson, The circular law for sparse non- H ermitian matrices , Ann. Probab. 47 (2019), no. 4, 2359--2416. 3980923

    2019

  17. [17]

    Theory Related Fields 159 (2014), no

    Paul Bourgade, Horng-Tzer Yau, and Jun Yin, The local circular law II : the edge case , Probab. Theory Related Fields 159 (2014), no. 3-4, 619--660. 3230004

    2014

  18. [18]

    Yuxin Chen, Chen Cheng, and Jianqing Fan, Asymmetry helps: eigenvalue and eigenvector analyses of asymmetrically perturbed low-rank matrices, Ann. Statist. 49 (2021), no. 1, 435--458. 4206685

    2021

  19. [19]

    Djalil Chafa\"i, Circular law for noncentral random matrices, J. Theoret. Probab. 23 (2010), no. 4, 945--950. 2735731

    2010

  20. [20]

    L\'aszl\'o Erd o s, Antti Knowles, Horng-Tzer Yau, and Jun Yin, Spectral statistics of E rd o s- R \'enyi G raphs II : E igenvalue spacing and the extreme eigenvalues , Comm. Math. Phys. 314 (2012), no. 3, 587--640. 2964770

    2012

  21. [21]

    L\'aszl\'o Erd o s, Antti Knowles, Horng-Tzer Yau, and Jun Yin, The local semicircle law for a general class of random matrices, Electron. J. Probab. 18 (2013), no. 59, 58. 3068390

    2013

  22. [22]

    L\'aszl\'o Erd o s, Antti Knowles, Horng-Tzer Yau, and Jun Yin, Spectral statistics of E rd o s- R \'enyi graphs I : L ocal semicircle law , Ann. Probab. 41 (2013), no. 3B, 2279--2375. 3098073

    2013

  23. [23]

    L\'aszl\'o Erd o s, Benjamin Schlein, and Horng-Tzer Yau, Local semicircle law and complete delocalization for W igner random matrices , Comm. Math. Phys. 287 (2009), no. 2, 641--655. 2481753

    2009

  24. [24]

    L\'aszl\'o Erd o s, Horng-Tzer Yau, and Jun Yin, Rigidity of eigenvalues of generalized W igner matrices , Adv. Math. 229 (2012), no. 3, 1435--1515. 2871147

    2012

  25. [25]

    4, 694--706

    Vyacheslav L Girko, Circular law, Theory of Probability & Its Applications 29 (1985), no. 4, 694--706

    1985

  26. [26]

    Friedrich G\"otze and Alexander Tikhomirov, The circular law for random matrices, Ann. Probab. 38 (2010), no. 4, 1444--1491. 2663633

    2010

  27. [27]

    Yukun He, Edge universality of sparse Erd o s - R \'e nyi digraphs , preprint (2025), arXiv:2304.04723 https://arxiv.org/abs/2304.04723

    work page internal anchor Pith review arXiv 2025

  28. [28]

    Yukun He, Antti Knowles, and Matteo Marcozzi, Local law and complete eigenvector delocalization for supercritical Erd o s - R \'e nyi graphs , Ann. Probab. 47 (2019), no. 5, 3278--3302

    2019

  29. [29]

    Multivariate Anal

    Guangming Pan and Wang Zhou, Circular law, extreme singular values and potential theory, J. Multivariate Anal. 101 (2010), no. 3, 645--656. 2575411

    2010

  30. [30]

    Mark Rudelson and Konstantin Tikhomirov, The sparse circular law under minimal assumptions, Geom. Funct. Anal. 29 (2019), no. 2, 561--637. 3945840

    2019

  31. [31]

    Walter Rudin, Real and complex analysis, third ed., McGraw-Hill Book Co., New York, 1987. 924157

    1987

  32. [32]

    Mark Rudelson and Roman Vershynin, Delocalization of eigenvectors of random matrices with independent entries, Duke Math. J. 164 (2015), no. 13, 2507--2538. 3405592

    2015

  33. [33]

    Theory Relat

    Terence Tao, Outliers in the spectrum of iid matrices with bounded rank perturbations, Probab. Theory Relat. Fields 155 (2013), no. 1-2, 231--263

    2013

  34. [34]

    Terence Tao and Van Vu, Random matrices: universality of ESD s and the circular law , Ann. Probab. 38 (2010), no. 5, 2023--2065, With an appendix by Manjunath Krishnapur. 2722794

    2010

  35. [35]

    3, 517--605

    Konstantin Tikhomirov and Pierre Youssef, Outliers in spectrum of sparse W igner matrices , Random Structures Algorithms 58 (2021), no. 3, 517--605. 4234995

    2021

  36. [36]

    Philip Matchett Wood, Universality and the circular law for sparse random matrices, Ann. Appl. Probab. 22 (2012), no. 3, 1266--1300. 2977992

    2012

This paper was first reviewed by grok-4.3 on June 25, 2026.