- The Oxford Handbook of Random Matrix Theory
- Dedication
- Foreword
- Detailed Contents
- List of Contributors
- Introduction and guide to the handbook
- History – an overview
- Symmetry classes
- Spectral statistics of unitary ensembles
- Spectral statistics of orthogonal and symplectic ensembles
- Universality
- Supersymmetry
- Replica approach in random matrix theory
- Painlevé transcendents
- Random matrix theory and integrable systems
- Determinantal point processes
- Random matrix representations of critical statistics
- Heavy-tailed random matrices
- Phase transitions
- Two-matrix models and biorthogonal polynomials
- Chain of matrices, loop equations, and topological recursion
- Unitary integrals and related matrix models
- Non-Hermitian ensembles
- Characteristic polynomials
- Beta ensembles
- Wigner matrices
- Free probability theory
- Random banded and sparse matrices
- Number theory
- Random permutations and related topics
- Enumeration of maps
- Knot theory and matrix integrals
- Multivariate statistics
- Algebraic geometry and matrix models
- Two-dimensional quantum gravity
- String theory
- Quantum chromodynamics
- Quantum chaos and quantum graphs
- Resonance scattering of waves in chaotic systems
- Condensed matter physics
- Classical and quantum optics
- Extreme eigenvalues of Wishart matrices: application to entangled bipartite system
- Random growth models
- Random matrices and Laplacian growth
- Financial applications of random matrix theory: a short review
- Asymptotic singular value distributions in information theory
- Random matrix theory and ribonucleic acid (RNA) folding
- Complex networks
- Index

## Abstract and Keywords

This article focuses on chiral random matrix theories with the global symmetries of quantum chromodynamics (QCD). In particular, it explains how random matrix theory (RMT) can be applied to the spectra of the Dirac operator both at zero chemical potential, when the Dirac operator is Hermitian, and at non-zero chemical potential, when the Dirac operator is non-Hermitian. Before discussing the spectra of these Dirac operators at non-zero chemical potential, the article considers spontaneous symmetry breaking in RMT and the QCD partition function. It then examines the global symmetries of QCD, taking into account the Dirac operator for a finite chiral basis, as well as the global symmetry breaking pattern and the Goldstone manifold in chiral random matrix theory (chRMT). It also describes the generating function for the Dirac spectrum and applications of chRMT to QCD to gauge degrees of freedom.

Keywords: quantum chromodynamics (QCD), random matrix theory (RMT), Dirac operator, zero chemical potential, non-zero chemical potential, spontaneous symmetry breaking, partition function, chiral random matrix theory, Dirac spectrum, degrees of freedom

Jacobus J. M. Verbaarschot, Department of Physics and Astronomy, State University of New York, Stony Brook, NY 11794, USA, verbaarschot@cs.physics.sunysb.edu

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- The Oxford Handbook of Random Matrix Theory
- Dedication
- Foreword
- Detailed Contents
- List of Contributors
- Introduction and guide to the handbook
- History – an overview
- Symmetry classes
- Spectral statistics of unitary ensembles
- Spectral statistics of orthogonal and symplectic ensembles
- Universality
- Supersymmetry
- Replica approach in random matrix theory
- Painlevé transcendents
- Random matrix theory and integrable systems
- Determinantal point processes
- Random matrix representations of critical statistics
- Heavy-tailed random matrices
- Phase transitions
- Two-matrix models and biorthogonal polynomials
- Chain of matrices, loop equations, and topological recursion
- Unitary integrals and related matrix models
- Non-Hermitian ensembles
- Characteristic polynomials
- Beta ensembles
- Wigner matrices
- Free probability theory
- Random banded and sparse matrices
- Number theory
- Random permutations and related topics
- Enumeration of maps
- Knot theory and matrix integrals
- Multivariate statistics
- Algebraic geometry and matrix models
- Two-dimensional quantum gravity
- String theory
- Quantum chromodynamics
- Quantum chaos and quantum graphs
- Resonance scattering of waves in chaotic systems
- Condensed matter physics
- Classical and quantum optics
- Extreme eigenvalues of Wishart matrices: application to entangled bipartite system
- Random growth models
- Random matrices and Laplacian growth
- Financial applications of random matrix theory: a short review
- Asymptotic singular value distributions in information theory
- Random matrix theory and ribonucleic acid (RNA) folding
- Complex networks
- Index