Lecture notes on random matrix theory
                      Charles Bordenave
                       January 11, 2019
         Foreword
         Some history
         Wishart, Von Neumann and Goldstine, Wigner, Dyson, Pastur, Girko, Voiculescu, ...
         Monographs
         Here are recent monographs on different topics in random matrix theory.
         [Tao12]: Terence Tao. Topics in random matrix theory, volume 132 of Graduate Studies in Math-
         ematics. American Mathematical Society, Providence, RI, 2012.
         [AGZ10]: Greg W. Anderson, Alice Guionnet, and Ofer Zeitouni. An introduction to random
         matrices, volume 118 of Cambridge Studies in Advanced Mathematics. Cambridge University
         Press, Cambridge, 2010.
         [BS10]: Zhidong Bai and Jack W. Silverstein. Spectral analysis of large dimensional random
         matrices. Springer Series in Statistics. Springer, New York, second edition, 2010.
         [EY17]: L´aszl´o Erd˝os and Horng-Tzer Yau. A dynamical approach to random matrix theory,
         volume 28 of Courant Lecture Notes in Mathematics. Courant Institute of Mathematical Sciences,
         New York; American Mathematical Society, Providence, RI, 2017.
         [For10]: P. J. Forrester. Log-gases and random matrices, volume 34 of London Mathematical
         Society Monographs Series. Princeton University Press, Princeton, NJ, 2010.
         [PS11]: Leonid Pastur and Mariya Shcherbina. Eigenvalue distribution of large random matrices,
         volume 171 of Mathematical Surveys and Monographs. American Mathematical Society, Provi-
         dence, RI, 2011.
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          Notation
          The vector space of n×n matrices on the field K ∈ {R,C} is denoted by M (K). The vector space
                                                        n
          of hermitian matrices is denoted by Hn(K).
            Wedenote by P and E the probability and the expectation of our underlying random variables.
                                        2
                Lecture 1
                Combinatorial proof of Wigner’s
                semicircle law
                1     Wigner’s semicircle Theorem
                1.1    Empirical distribution of eigenvalues
                Let X be an hermitian matrix in M (C). Counting multiplicities, its ordered eigenvalues are
                                                        n
                denoted as
                                                        λ1(X) ≥ ··· ≥ λn(X).
                    The empirical spectral distribution (ESD) is
                                                                    n
                                                          µX = 1 Xδλ(X).
                                                                 n       i
                                                                   i=1
                    This is a global function of the spectrum. From the spectral theorem, for any function f,
                                                     Z f(λ)dµ (λ) = 1Trf(X).
                                                               X        n
                1.2    Wigner matrix
                Consider an infinite array of complex random variables (Xij) where for 1 ≤ i < j
                                                                      ¯
                                                              X =X
                                                                ij     ji
                are iid with law P on C, independent of Xii,i ≥ 1 iid with common law Q on R.
                    The random matrix X = (X )                is hermitian. This matrix is called a Wigner matrix.
                                                   ij 1≤i,j≤n
                There are some important cases:
                                                                   3