Extremal Laws for Laplacian Random Matrices

TL;DR

This paper proves that for Gaussian Laplacian random matrices, the fluctuations of the largest eigenvalue and diagonal entry follow a Gumbel distribution, and discusses spectral properties and extensions.

Contribution

It establishes non-asymptotic bounds for the largest eigenvalue and compares Laplacian matrix spectra to Wigner matrices, highlighting differences and extensions.

Findings

Largest eigenvalue fluctuations are Gumbel distributed.

Provides bounds relating largest eigenvalue to diagonal entries.

Reviews spectral properties of Laplacian matrices.

Abstract

For an $n\times n$ Laplacian random matrix $L$ with Gaussian entries it is proven that the fluctuations of the largest eigenvalue and the largest diagonal entry of $L/\sqrt{n-1}$ are Gumbel. We first establish suitable non-asymptotic estimates and bounds for the largest eigenvalue of $L$ in terms of the largest diagonal element of $L$. An expository review of existing results for the asymptotic spectrum of a Laplacian random matrix is also presented, with the goal of noting the differences from the corresponding classical results for Wigner random matrices. Extensions to Laplacian block random matrices are indicated.