An introduction to random matrices

Luca Guido Molinari

(preliminary drafts of notes)

Lecture Notes

LESSON 1: Beautiful theorems. Some special matrices. Distributions of eigenvalues of large random matrices (april 2018, revised JULY 2021)

LESSON 2: Hermitian 1-matrix models (From matrix elements to eigenvalues. Stieltjes, saddle point, orthogonal polynomials. GUE. Sine and Airy correlators. Tracy Widom. Topological expansion and diagram counting. (april 2018, revised april 2019).

LESSON 3: The Laplace-Beltrami operator for Hermitian matrices, Harish-Chandra formula (proofs by Brezin and Mehta), the 2-matrix model (bi-orthogonal polynomials, phase transition), 1-matrix model D=1 (may 2019).

LESSON 4: Ising model on planar graphs (may 2019)

LESSON: Tridiagonal matrices.

LESSON 5: Random surfaces, 2d quantum gravity, and random matrices.

Seminari degli allievi di dottorato, tesi di laurea

Tommaso Rossi: Teoremi di localizzazione spettrale. (laurea triennale, a.a. 2016/17)
Giovanni Stagnitto: Introduzione alla misura di Haar (7 march 2019)
Federico Faedo: Random surfaces applied (10 october 2019)
Vittorio Erba: Notes on the enumeration of RNA secondary structures by matrix models (23 october 2019)
Mauro Pastore: The QCD partition function and Chiral Random Matrices (28 march 2019)
Martina Toscani: How to use random matrix theory in the detection of gravitational waves (1 july 2020)

Random Matrices: Theory and Applications (a journal on RM)
Brunel-Bielefeld Workshop Random Matrix Theory and Applications

Books and lecture notes:
G.Akemann, J.Baik and Ph.Di Francesco, The Oxford Handbook of random matrix theory, Oxford Univ Press 2011.
Madan Lal Mehta, Random Matrices Eds. 1, 2, 3
Peter J. Forrester, Log-Gases and Random Matrices, London Math. Soc. Monographs, Princeton University Press 2010.
Leonid Pastur and Mariya Shcherbina, Eigenvalues of Large Random Matrices, Mathematical Surveys and Monographs 171, AMS 2011.
Edouard Brezin and Shinobu Hikami, Random Matrix Theory with and external source, Springer 2016.
Giacomo Livan, Marcel Novaes, Pierpaolo Vivo, Introduction to Random Matrices - Theory and Practice (arXiv:1712.07903). Printed by Springer
Luis Alvarez Gaume`, Random surfaces, statistical mechanics and string theory (Helvetica Physica Acta 64 n.4 (1991) 359-526).
Pavel M. Bleher, Lectures on random matrix models. The Riemann Hilbert approach 84 pages (arxiv:0801.1858).
Terence Tao, Topics in random matrix theory (2012, printed by AMS).

(from 10 july 2020)