Universality for the extreme eigenvalues of Laplacian random matrices

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 Erd\H{o}s--R\'enyi random graph, we show the asymptotic fluctuations of the algebraic connectivity (Fiedler value) can be described in terms of the Gumbel distribution.