Random matrices
Mathematical Physics
2018-07-06 v2 Statistical Mechanics
High Energy Physics - Theory
math.MP
Abstract
We provide a self-contained introduction to random matrices. While some applications are mentioned, our main emphasis is on three different approaches to random matrix models: the Coulomb gas method and its interpretation in terms of algebraic geometry, loop equations and their solution using topological recursion, orthogonal polynomials and their relation with integrable systems. Each approach provides its own definition of the spectral curve, a geometric object which encodes all the properties of a model. We also introduce the two peripheral subjects of counting polygonal surfaces, and computing angular integrals.
Cite
@article{arxiv.1510.04430,
title = {Random matrices},
author = {Bertrand Eynard and Taro Kimura and Sylvain Ribault},
journal= {arXiv preprint arXiv:1510.04430},
year = {2018}
}
Comments
196 pages, v2: major revision and expansion, 32 exercises added