Large deviations for macroscopic observables of heavy-tailed matrices

Charles Bordenave; Alice Guionnet; Camille Male

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arXiv · 2409.14027

Large deviations for macroscopic observables of heavy-tailed matrices

Charles Bordenave·Alice Guionnet·Camille Male

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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.

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Charles Bordenave, Alice Guionnet, Camille Male. 2024-09-21. Large deviations for macroscopic observables of heavy-tailed matrices. https://arxiv.org/abs/2409.14027

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