Extreme eigenvalues of Laplacian random matrices with Gaussian entries

TL;DR

This paper studies the extreme eigenvalues of Gaussian Laplacian matrices, revealing they follow Gumbel distribution and exhibit Poisson statistics, with surprising differences in centering terms.

Contribution

It establishes the limiting distribution of the largest eigenvalues of Laplacian Gaussian matrices, a model with dependent entries, showing Gumbel fluctuations and Poisson statistics.

Findings

Largest eigenvalues follow Gumbel distribution

Eigenvalues exhibit Poisson statistics in the limit

Largest diagonal entry also has Gumbel fluctuations

Abstract

A Laplacian matrix is a real symmetric matrix whose row and column sums are zero. We investigate the limiting distribution of the largest eigenvalues of a Laplacian random matrix with Gaussian entries. Unlike many classical matrix ensembles, this random matrix model contains dependent entries. Our main results show that the extreme eigenvalues of this model exhibit Poisson statistics. In particular, after properly shifting and scaling, we show that the largest eigenvalue converges to the Gumbel distribution as the dimension of the matrix tends to infinity. While the largest diagonal entry is also shown to have Gumbel fluctuations, there is a rather surprising difference between its deterministic centering term and the centering term required for the largest eigenvalues.