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Universality for the extreme eigenvalues of Laplacian random matrices

  • Andrew Campbell
  • Kyle Luh
  • Sean O'Rourke
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Abstract

We study the eigenvalues of the random Laplacian matrix $D - A$, where A is a Wigner matrix with sub-Gaussian entries and the diagonal matrix D contains the row sums of A. Our main results show that the extreme eigenvalues of this model exhibit Poisson statistics; in particular, after the appropriate centering and scaling, the largest eigenvalue converges to the Gumbel distribution as the dimension of the matrix tends to infinity. This confirms, for general sub-Gaussian entries, a phenomenon the authors previously established only in the Gaussian case [Electron. J. Probab. 30 (2025), Paper No. 104], resolving a conjecture raised there. As a corollary, for an Erdős--Rényi random graph, we show the asymptotic fluctuations of the algebraic connectivity (Fiedler value) can be described in terms of the Gumbel distribution.

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Cite

BibTeXRIS
BibTeX
@misc{campbell2026universality,
  title = {{Universality for the extreme eigenvalues of Laplacian random matrices}},
  author = {Andrew Campbell and Kyle Luh and Sean O'Rourke},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.09076},
  eprint = {2610.09076},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.09076},
}
RIS
TY  - UNPB
TI  - Universality for the extreme eigenvalues of Laplacian random matrices
AU  - Andrew Campbell
AU  - Kyle Luh
AU  - Sean O'Rourke
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.09076
UR  - https://arxiv.org/abs/2610.09076
AB  - We study the eigenvalues of the random Laplacian matrix $D - A$, where A is a Wigner matrix with sub-Gaussian entries and the diagonal matrix D contains the row sums of A. Our main results show that the extreme eigenvalues of this model exhibit Poisson statistics; in particular, after the appropriate centering and scaling, the largest eigenvalue converges to the Gumbel distribution as the dimension of the matrix tends to infinity. This confirms, for general sub-Gaussian entries, a phenomenon the authors previously established only in the Gaussian case [Electron. J. Probab. 30 (2025), Paper No. 104], resolving a conjecture raised there. As a corollary, for an Erdős--Rényi random graph, we show the asymptotic fluctuations of the algebraic connectivity (Fiedler value) can be described in terms of the Gumbel distribution.
ER  -