An introduction to random matrices
Luca Guido Molinari
(preliminary drafts of notes)
Lecture Notes
LESSON 1: Beautiful theorems.
(Beautiful theorems, some special matrices. Distributions of
eigenvalues of large random matrices) (april 2018, revised feb 2023)
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: Haar Measures of classical groups
(Euler angles for SO(n), invariant measures of GL(n,R) and GL(n,C), Haar
measures on classical groups: Euler angles, Weyl parametrization, Cayley transform)
LESSON 5: Ising model on planar graphs (may 2019)
LESSON 6: Tridiagonal matrices.
(Transfer matrix, Christoffel-Darboux sums, periodic matrices, models of random matrices, localization, Thouless formula, Lanczos and Householder,
dynamical localization)
LESSON 7: Random surfaces, 2d quantum gravity, and random matrices.
(Euler number and Gauss-Bonnet theorem, String theory and random surfaces, triangulations)
LESSON 8: Positive matrices and statistics
(Positive matrices, parametrizations and Haar measures.
Integrals with positive matrix variable. Real rectangular matrices.
Wishart distribution. Laguerre ensemble and Marchenko-Pastur law.
Spherical distributions)
Seminars of doctorate students, laurea thesis
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).
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(from 10 july 2020)