Extreme value problems in Random Matrix Theory and other disordered systems

TL;DR

This paper reviews the application of extreme value statistics and central limit theorems in disordered systems, highlighting the impact of fat-tailed distributions on Random Matrix Theory and related models.

Contribution

It discusses the generalization of Tracy-Widom distribution for fat-tailed disorders and emphasizes the importance of non-Gaussian assumptions in disordered systems.

Findings

Fat tails significantly affect eigenvalue distributions.

Generalization of Tracy-Widom distribution for non-Gaussian disorders.

Open problems in the theory of disordered systems with power-law tails.

Abstract

We review some applications of central limit theorems and extreme values statistics in the context of disordered systems. We discuss several problems, in particular concerning Random Matrix Theory and the generalisation of the Tracy-Widom distribution when the disorder has ``fat tails''. We underline the relevance of power-law tails for Directed Polymers and mean-field Spin Glasses, and we point out various open problems and conjectures on these matters. We find that in many instances the assumption of Gaussian disorder cannot be taken for granted.