Spectrum of Laplacian matrices associated with large random elliptic matrices

TL;DR

This paper investigates the eigenvalue distribution of Laplacian matrices derived from large random elliptic matrices, revealing convergence to a deterministic measure linked to free probability theory.

Contribution

It introduces the limiting spectral distribution of Laplacian matrices from elliptic matrices, connecting random matrix theory with free probability.

Findings

Empirical spectral distribution converges to a deterministic measure.

The limiting measure relates to the Brown measure of a sum of operators.

Provides insights into the spectral properties of large random elliptic matrices.

Abstract

A Laplacian matrix is a square matrix whose row sums are zero. We study the limiting eigenvalue distribution of a Laplacian matrix formed by taking a random elliptic matrix and subtracting the diagonal matrix containing its row sums. Under some mild assumptions, we show that the empirical spectral distribution of the Laplacian matrix converges to a deterministic probability distribution as the size of the matrix tends to infinity. The limiting measure can be interpreted as the Brown measure of the sum of an elliptic operator and a freely independent normal operator with a Gaussian distribution.