Random matrices

Bertrand Eynard (1) , Taro Kimura (2, 1) , Sylvain Ribault (1)

1 IPHT - Institut de Physique Théorique - UMR CNRS 3681
2 RIKEN - RIKEN - Institute of Physical and Chemical Research [Japon]

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.

Keywords: Random matrix; eigenvalue distribution; ribbon graph; large size limit; random surface; spectral curve; loop equations; topological recursion; orthogonal polynomials; integrable systems 1

Domains: Physics [physics]; Mathematical Physics [math-ph]; Physics [physics]; High Energy Physics - Theory [hep-th]

Bertrand Eynard, Taro Kimura, Sylvain Ribault. Random matrices. 2016. ⟨cea-01252029⟩