# On the extreme value statistics of normal random matrices and 2D Coulomb gases: Universality and finite N corrections

## TL;DR

This paper extends orthogonal polynomial methods to analyze extreme eigenvalue statistics of normal random matrices and 2D Coulomb gases, demonstrating universality, finite N corrections, and connections to Hermitian matrix techniques.

## Contribution

It introduces a generalized approach for extreme value analysis of non-Hermitian matrices, including finite N corrections and universality results, using methods adapted from Hermitian matrix theory.

## Key findings

- Eigenvalue with largest modulus converges to Gumbel distribution
- Universality holds for arbitrary radially symmetric potentials
- Finite N corrections are explicitly computed

## Abstract

In this paper we extend the orthogonal polynomials approach for extreme value calculations of Hermitian random matrices, developed by Nadal and Majumdar [1102.0738], to normal random matrices and 2D Coulomb gases in general. Firstly, we show that this approach provides an alternative derivation of results in the literature. More precisely, we show convergence of the rescaled eigenvalue with largest modulus of a normal Gaussian ensemble to a Gumbel distribution, as well as universality for an arbitrary radially symmetric potential. Secondly, it is shown that this approach can be generalised to obtain convergence of the eigenvalue with smallest modulus and its universality for ring distributions. Most interestingly, the here presented techniques are used to compute all slowly varying finite N correction of the above distributions, which is important for practical applications, given the slow convergence. Another interesting aspect of this work is the fact that we can use standard techniques from Hermitian random matrices to obtain the extreme value statistics of non-Hermitian random matrices resembling the large N expansion used in context of the double scaling limit of Hermitian matrix models in string theory.