Random Matrix Theory and Quantum Chromodynamics

TL;DR

This paper explores the application of random matrix theory to Quantum Chromodynamics, focusing on eigenvalue distributions, symmetries, and recent developments like finite chemical potential effects.

Contribution

It provides a detailed analysis of the chiral Gaussian Unitary Ensemble and connects it to QCD spectra, including recent advances and open problems.

Findings

Eigenvalue density correlation functions derived

Microscopic limit at the origin analyzed

Effects of finite chemical potential discussed

Abstract

These notes are based on the lectures delivered at the Les Houches Summer School in July 2015. They are addressed at a mixed audience of physicists and mathematicians with some basic working knowledge of random matrix theory. The first part is devoted to the solution of the chiral Gaussian Unitary Ensemble in the presence of characteristic polynomials, using orthogonal polynomial techniques. This includes all eigenvalue density correlation functions, smallest eigenvalue distributions and their microscopic limit at the origin. These quantities are relevant for the description of the Dirac operator spectrum in Quantum Chromodynamics with three colours in four Euclidean space-time dimensions. In the second part these two theories are related based on symmetries, and the random matrix approximation is explained. In the last part recent developments are covered including the effect of finite chemical potential and finite space-time lattice spacing, and their corresponding orthogonal polynomials. We also give some open random matrix problems.