viii CONTENTS 2.4.1 Gaussian Wigner matrices 46 2.4.2 General Wigner matrices 47 2.5 Joint distribution of eigenvalues in the GOE and the GUE 51 2.5.1 Definition and preliminary discussion of the GOE and the GUE 51 2.5.2 Proof of the joint distribution of eigenvalues 54 2.5.3 Selberg's integral formula and proof of(2.5.4) 59 2.5.4 Joint distribution of eigenvalues-alternative formu- lation 65 2.5.5 Superposition and decimation relations 66 2.6 Large deviations for random matrices 71 2.6.1 Large deviations for the empirical measure 72 2.6.2 Large deviations for the top eigenvalue 82 2.7 Bibliographical notes 86 w Hermite polynomials,spacings,and limit distributions for the Gaus- sian ensembles 91 3.1 Summary of main results:spacing distributions in the bulk and edge of the spectrum for the Gaussian ensembles 91 3.1.1 Limit results for the GUE 91 3.1.2 Generalizations:limit formulas for the GOE and GSE 94 3.2 Hermite polynomials and the GUE 95 3.2.1 The GUE and determinantal laws 95 3.2.2 Properties of the Hermite polynomials and oscillator wave-functions 100 3.3 The semicircle law revisited 103 3.3.1 Calculation of moments of Ly 103 3.3.2 The Harer-Zagier recursion and Ledoux's argument 105 3.4 Quick introduction to Fredholm determinants 108 3.4.1 The setting,fundamental estimates,and definition of the Fredholm determinant 108 3.4.2 Definition of the Fredholm adjugant,Fredholm resolvent,and a fundamental identity 111viii CONTENTS 2.4.1 Gaussian Wigner matrices 46 2.4.2 General Wigner matrices 47 2.5 Joint distribution of eigenvalues in the GOE and the GUE 51 2.5.1 Definition and preliminary discussion of the GOE and the GUE 51 2.5.2 Proof of the joint distribution of eigenvalues 54 2.5.3 Selberg’s integral formula and proof of (2.5.4) 59 2.5.4 Joint distribution of eigenvalues - alternative formu￾lation 65 2.5.5 Superposition and decimation relations 66 2.6 Large deviations for random matrices 71 2.6.1 Large deviations for the empirical measure 72 2.6.2 Large deviations for the top eigenvalue 82 2.7 Bibliographical notes 86 3 Hermite polynomials, spacings, and limit distributions for the Gaus￾sian ensembles 91 3.1 Summary of main results: spacing distributions in the bulk and edge of the spectrum for the Gaussian ensembles 91 3.1.1 Limit results for the GUE 91 3.1.2 Generalizations: limit formulas for the GOE and GSE 94 3.2 Hermite polynomials and the GUE 95 3.2.1 The GUE and determinantal laws 95 3.2.2 Properties of the Hermite polynomials and oscillator wave-functions 100 3.3 The semicircle law revisited 103 3.3.1 Calculation of moments of L¯N 103 3.3.2 The Harer–Zagier recursion and Ledoux’s argument 105 3.4 Quick introduction to Fredholm determinants 108 3.4.1 The setting, fundamental estimates, and definition of the Fredholm determinant 108 3.4.2 Definition of the Fredholm adjugant, Fredholm resolvent, and a fundamental identity 111