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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

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