Distribution functions for largest eigenvalues and their applications

TL;DR

This paper surveys the distribution functions of the largest eigenvalues in classic random matrix models, highlighting their universal limit laws and applications in physics and particle systems, expressed via Painlevé II functions.

Contribution

It provides a comprehensive overview of the distribution functions for largest eigenvalues and their universality across different models and applications.

Findings

Distribution functions are expressed in terms of Painlevé II functions.

Universal limit laws apply to a wide range of physical and mathematical systems.

Survey of occurrences in mathematical physics and interacting particle systems.

Abstract

It is now believed that the limiting distribution function of the largest eigenvalue in the three classic random matrix models GOE, GUE and GSE describe new universal limit laws for a wide variety of processes arising in mathematical physics and interacting particle systems. These distribution functions, expressed in terms of a certain Painlev\'e II function, are described and their occurences surveyed.