Introduction to the Random Matrix Theory: Gaussian Unitary Ensemble and Beyond

TL;DR

This paper introduces the fundamental concepts and tools of random matrix theory, focusing on the Gaussian Unitary Ensemble, orthogonal polynomial methods, and their spectral properties, with insights into current research directions.

Contribution

It provides an accessible overview of the analysis techniques for eigenvalues of large random Hermitian matrices, emphasizing GUE and orthogonal polynomial connections.

Findings

Asymptotic behavior of Hermite polynomials analyzed

Spectral properties of GUE discussed in detail

Connections between orthogonal and characteristic polynomials explored

Abstract

These lectures provide an informal introduction into the notions and tools used to analyze statistical properties of eigenvalues of large random Hermitian matrices. After developing the general machinery of orthogonal polynomial method, we study in most detail Gaussian Unitary Ensemble (GUE) as a paradigmatic example. In particular, we discuss Plancherel-Rotach asymptotics of Hermite polynomials in various regimes and employ it in spectral analysis of the GUE. In the last part of the course we discuss general relations between orthogonal polynomials and characteristic polynomials of random matrices which is an active area of current research.