CONTENTS 车 3.5 Gap probabilities at 0 and proof of Theorem 3.1.1. 116 3.5.1 The method of Laplace 117 3.5.2 Evaluation of the scaling limit-proof of Lemma 3.5.1 119 3.5.3 A complement:determinantal relations 122 3.6 Analysis of the sine-kernel 123 3.6.1 General differentiation formulas 123 3.6.2 Derivation of the differential equations:proof of Theorem 3.6.1 128 3.6.3 Reduction to Painleve V 130 3.7 Edge-scaling:Proof of Theorem 3.1.4 134 3.7.1 Vague convergence of the rescaled largest eigen- value:proofof Theorem 3.1.4 135 3.7.2 Steepest descent:proof of Lemma 3.7.2 136 3.7.3 Properties of the Airy functions and proofof Lemma 3.7.1 141 3.8 Analysis of the Tracy-Widom distribution and proof of Theorem 3.1.5 144 3.8.1 The first standard moves of the game 146 3.8.2 The wrinkle in the carpet 147 3.8.3 Linkage to Painleve II 148 3.9 Limiting behavior of the GOE and the GSE 150 3.9.1 Pfaffians and gap probabilities 150 3.9.2 Fredholm representation of gap probabilities 158 3.9.3 Limit calculations 163 3.9.4 Differential equations 172 3.10 Bibliographical notes 183 4 Some generalities 188 4.1 Joint distribution of eigenvalues in the classical matrix ensembles 189 4.1.1 Integration formulas for classical ensembles 189 4.1.2 Manifolds,volume measures,and the coarea formula 195CONTENTS ix 3.5 Gap probabilities at 0 and proof of Theorem 3.1.1. 116 3.5.1 The method of Laplace 117 3.5.2 Evaluation of the scaling limit – proof of Lemma 3.5.1 119 3.5.3 A complement: determinantal relations 122 3.6 Analysis of the sine-kernel 123 3.6.1 General differentiation formulas 123 3.6.2 Derivation of the differential equations: proof of Theorem 3.6.1 128 3.6.3 Reduction to Painlev´e V 130 3.7 Edge-scaling: Proof of Theorem 3.1.4 134 3.7.1 Vague convergence of the rescaled largest eigen￾value: proof of Theorem 3.1.4 135 3.7.2 Steepest descent: proof of Lemma 3.7.2 136 3.7.3 Properties of the Airy functions and proof of Lemma 3.7.1 141 3.8 Analysis of the Tracy-Widom distribution and proof of Theorem 3.1.5 144 3.8.1 The first standard moves of the game 146 3.8.2 The wrinkle in the carpet 147 3.8.3 Linkage to Painlev´e II 148 3.9 Limiting behavior of the GOE and the GSE 150 3.9.1 Pfaffians and gap probabilities 150 3.9.2 Fredholm representation of gap probabilities 158 3.9.3 Limit calculations 163 3.9.4 Differential equations 172 3.10 Bibliographical notes 183 4 Some generalities 188 4.1 Joint distribution of eigenvalues in the classical matrix ensembles 189 4.1.1 Integration formulas for classical ensembles 189 4.1.2 Manifolds, volume measures, and the coarea formula 195