Random Matrix Models

The lectures will be about the theory of random matrices (RMT). Broadly speaking, this theory deals with the statistical properties of a matrix of a large size. Originally introduced by Wigner to describe the energy levels of heavy nuclei, nowadays, RMT is used in various fields of physics and mathematics, ranging from quantum chaos and transport properties in solids to string theory and quantum gravity.

The aim of the lectures is to introduce students to the formalism of random matrices. In these lectures, we will

• learn about various ensembles of random matrices

• derive eigenvalue probability distributions (including famous Wigner semicircle law)

• develop the perturbative techniques, and learn about Feynman diagrams in the simplest possible setup

• find out the relation between the 2D plasmas with logarithmic interactions and Quantum Hall effects, and learn how to deal with these systems.

Besides interesting application to the physical problem, the course allows learning techniques broadly used in other branches of theoretical physics, particularly in quantum field theory (Feynman diagrams, Ward identities, etc.). If time permits, we will also study mathematical questions that appear in the course. In particular, it is planned to cover, at least on the basic level, the theory of orthogonal polynomials, Painleve equations, and integrable systems.

The plan of the lectures:

1. Random matrices in physics and mathematics.

2. Ensembles of Random matrices. Symmetries. Unitary ensembles.

3. Plasma picture. Large N solution for the eigenvalue distributions of the Hermitian matrix. (Loop equation, Wigner semicircle law)

4. Perturbative method: Feynman diagrams; triangulation of the 2D surfaces

5. Normal matrices and 2D support of the eigenvalues.

6. Orthogonal polynomial and integrable systems (if time permits)

1. Bertrand Eynard, Taro Kimura and Sylvain Ribault, "Random matrices", https://arxiv.org/abs/1510.04430

2. A. Zabrodin, "Matrix models and growth processes: from viscous flows to the quantum Hall effect", https://arxiv.org/abs/hep-th/0412219

3. C.W.J. Beenakker, "Random-Matrix Theory of Quantum Transport", https://arxiv.org/abs/cond-mat/9612179

4. Peter J. Forrester, "Log-Gases and Random Matrices", Princeton University Press 2010

Students will understand basic features of random matrices and their relations to various physical systems. They will be able to describe various ensembles in terms of the eigenvalue distributions. For the Hermitian and Normal Gaussian ensembles, they will find the corresponding distributions explicitly. Students will understand the structure of the generic models and will learn how to investigate it using perturbative methods, such as Feynmann diagrams. Learning these techniques is also extremely useful for other branches of modern physics.

Based on graded homework exercises and final exam.