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PreprintCC BY 4.0arXiv

The spectral edge of sparse directed Erdös-Rényi graphs

  • Simon Coste
  • Yizhe Zhu
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Abstract

Let d>1 be fixed and let An be an n× n matrix with independent Ber(d/n) entries. For every 0<r<√d, we prove that, with high probability, a positive proportion of the eigenvalues of An have modulus larger than r. Together with the known upper bound, this implies that the modulus of the second largest eigenvalue converges in probability to √d. Our proof works with the Brown measure μd of the adjacency operator of the directed Poisson--Galton--Watson tree. The convergence theorem of Sah, Sahasrabudhe, and Sawhney, together with the Brown-measure identification following it, gives weak convergence of the empirical spectral measure of An to μd in probability. We prove that the outer radius of its support is √d, using a resolvent recursion on Poisson--Galton--Watson trees.

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Cite

BibTeXRIS
BibTeX
@misc{coste2026spectral,
  title = {{The spectral edge of sparse directed Erdös-Rényi graphs}},
  author = {Simon Coste and Yizhe Zhu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.18369},
  eprint = {2609.18369},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18369},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The spectral edge of sparse directed Erdös-Rényi graphs
AU  - Simon Coste
AU  - Yizhe Zhu
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.18369
UR  - https://arxiv.org/abs/2609.18369
AB  - Let d>1 be fixed and let A_n be an n× n matrix with independent Ber(d/n) entries. For every 0<r<√d, we prove that, with high probability, a positive proportion of the eigenvalues of A_n have modulus larger than r. Together with the known upper bound, this implies that the modulus of the second largest eigenvalue converges in probability to √d. Our proof works with the Brown measure μ_d of the adjacency operator of the directed Poisson--Galton--Watson tree. The convergence theorem of Sah, Sahasrabudhe, and Sawhney, together with the Brown-measure identification following it, gives weak convergence of the empirical spectral measure of A_n to μ_d in probability. We prove that the outer radius of its support is √d, using a resolvent recursion on Poisson--Galton--Watson trees.
ER  -