This course is an introduction to the theory of random matrices, one of the most active research topics in contemporary mathematical physics and probability. In addition to its intrinsic mathematical appeal, interest in random matrices has been spurred by the scientific hypothesis that large random matrices yield models for complex systems comprised of many highly correlated components. Such systems are ubiquitous in mathematics and nature (energy levels of heavy nuclei or chaotic quantum billiards, zeros of L-functions, random growth models, etc.) but are not within the purview of classical scalar probability theory, whose limit theorems usually apply to systems of weakly correlated components. Topics covered will include: brief history of random matrix theory; basic objects and questions; the main limit theorems; connections to other areas of mathematics and science; classical matrix models (Gaussian and unitary); semicircular law; determinantal point processes, orthogonal polynomials and scaling limits; gap probabilities; statistics of the largest eigenvalue and Tracy-Widom distributions; log-gas and the equilibrium measure; non-hermitian random matrices.
- Prerequisites: basic linear algebra and probability.
- References:
- M. L. Mehta, Random Matrices, 1967.
- G. W. Anderson, A. Guionnet, O. Zeitouni, An introduction to Random Matrices, 2005.
- P. J. Forrester, Log-gases and Random Matrices, 2010.
- T. Tao, Topics in Random Matrix Theory, 2012.
- G. Livan, M. Novaes, P. Vivo, Introduction to Random Matrices-Theory and Practice, 2018.
