Random Matrix Theory and Quantum Chromodynamics

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.

Les Houches lecture notes, Session July 2015, 37 pages, 6 figures, v2: typos corrected and grant no. added, version to appear

References in corpus (7)

• Recent exact and asymptotic results for products of independent random matrices
• Random matrix analysis of the QCD sign problem for general topology
• Dropping the independence: singular values for products of two coupled random matrices
• Subsets and the canonical partition functions
• Finite size scaling of meson propagators with isospin chemical potential
• Dirac Spectra of 2-dimensional QCD-like theories
• The microscopic Twisted Mass Dirac spectrum and the spectral determination of the LECs of Wilson $\chi$-PT

Cited by in corpus (9)

• Cornering Large-$N_c$ QCD with Positivity Bounds
• Many-body quantum chaos in stroboscopically-driven cold atoms
• Chiral Symmetry Breaking on the Lattice
• Distribution of the ratio of two consecutive level spacings in orthogonal to unitary crossover ensembles
• Metal or Insulator? Dirac operator spectrum in holographic QCD
• Stochastic calculation of the Dirac spectrum on the lattice and a determination of chiral condensate in 2+1-flavor QCD
• Spectral Statistics of Non-Hermitian Matrices and Dissipative Quantum Chaos
• Universal microscopic spectrum of the unquenched QCD Dirac operator at finite temperature
• Hermitian and non-Hermitian perturbations of chiral Gaussian $\beta$-ensembles