Contents Preface page xiii 1 Introduction 1 2 Real and Complex Wigner matrices 6 2.1 Real Wigner matrices:traces,moments and combinatorics 6 2.1.1 The semicircle distribution,Catalan numbers,and Dyck paths 1 2.1.2 Proof#1 of Wigner's Theorem 2.1.1 10 2.1.3 Proof of Lemma 2.1.6:Words and Graphs 11 2.1.4 Proof of Lemma 2.1.7:Sentences and Graphs 17 2.1.5 Some useful approximations 21 2.1.6 Maximal eigenvalues and Furedi-Komlos enumeration 23 2.1.7 Central limit theorems for moments 29 2.2 Complex Wigner matrices 35 2.3 Concentration for functionals of random matrices and logarithmic Sobolev inequalities 38 2.3.1 Smoothness properties of linear functions of the empirical measure 38 2.3.2 Concentration inequalities for independent variables satisfying logarithmic Sobolev inequalities 39 2.3.3 Concentration for Wigner-type matrices 42 2.4 Stieltjes transforms and recursions 43 viiContents Preface page xiii 1 Introduction 1 2 Real and Complex Wigner matrices 6 2.1 Real Wigner matrices: traces, moments and combinatorics 6 2.1.1 The semicircle distribution, Catalan numbers, and Dyck paths 7 2.1.2 Proof #1 of Wigner’s Theorem 2.1.1 10 2.1.3 Proof of Lemma 2.1.6 : Words and Graphs 11 2.1.4 Proof of Lemma 2.1.7 : Sentences and Graphs 17 2.1.5 Some useful approximations 21 2.1.6 Maximal eigenvalues and F¨uredi-Koml´os enumeration 23 2.1.7 Central limit theorems for moments 29 2.2 Complex Wigner matrices 35 2.3 Concentration for functionals of random matrices and logarithmic Sobolev inequalities 38 2.3.1 Smoothness properties of linear functions of the empirical measure 38 2.3.2 Concentration inequalities for independent variables satisfying logarithmic Sobolev inequalities 39 2.3.3 Concentration for Wigner-type matrices 42 2.4 Stieltjes transforms and recursions 43 vii