Large deviations for macroscopic observables of heavy-tailed matrices

TL;DR

This paper establishes a large deviations principle for macroscopic observables of heavy-tailed random matrices, including eigenvalue distributions and traffic measures, using graph representations and entropy concepts.

Contribution

It introduces a large deviations framework for heavy-tailed matrices, encompassing Levy and sparse models, and connects traffic distributions with free convolution entropy.

Findings

Large deviations principle for eigenvalue and traffic distributions.

Representation of matrices as weighted graphs for analysis.

Definition of microstates entropy additive under free traffic convolution.

Abstract

We consider a finite collection of independent Hermitian heavy-tailed random matrices of growing dimension. Our model includes the L\'evy matrices proposed by Bouchaud and Cizeau, as well as sparse random matrices with O(1) non-zero entries per row. By representing these matrices as weighted graphs, we derive a large deviations principle for key macroscopic observables. Specifically, we focus on the empirical distribution of eigenvalues, the joint neighborhood distribution, and the joint traffic distribution. As an application, we define a notion of microstates entropy for traffic distributions which is additive for free traffic convolution.