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Preface | |

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Introduction | |

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Real and complex Wigner matrices | |

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Real Wigner matrices: traces, moments and combinatorics | |

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The semicircle distribution, Catalan numbers and Dyck paths | |

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Proof #1 of Wigner's Theorem 2.1.1 | |

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Proof of Lemma 2.1.6: words and graphs | |

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Proof of Lemma 2.1.7: sentences and graphs | |

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Some useful approximations | |

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Maximal eigenvalues and Fï¿½redi-Komlï¿½s enumeration | |

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Central limit theorems for moments | |

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Complex Wigner matrices | |

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Concentration for functionals of random matrices and logarithmic Sobolev inequalities | |

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Smoothness properties of linear functions of the empirical measure | |

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Concentration inequalities for independent variables satisfying logarithmic Sobolev inequalities | |

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Concentration for Wigner-type matrices | |

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Stieltjes transforms and recursions | |

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Gaussian Wigner matrices | |

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General Wigner matrices | |

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Joint distribution of eigenvalues in the GOE and the GUE | |

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Definition and preliminary discussion of the GOE and the GUE | |

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Proof of the joint distribution of eigenvalues | |

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Selberg's integral formula and proof of (2.5.4) | |

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Joint distribution of eigenvalues: alternative formulation | |

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Superposition and decimation relations | |

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Large deviations for random matrices | |

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Large deviations for the empirical measure | |

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Large deviations for the top eigenvalue | |

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Bibliographical notes | |

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Hermite polynomials, spacings and limit distributions for the Gaussian ensembles | |

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Summary of main results: spacing distributions in the bulk and edge of the spectrum for the Gaussian ensembles | |

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Limit results for the GUE | |

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Generalizations: limit formulas for the GOE and GSE | |

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Hermite polynomials and the GUE | |

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The GUE and determinantal laws | |

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Properties of the Hermite polynomials and oscillator wave-functions | |

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The semicircle law revisited | |

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Calculation of moments of L<sub>N</sub> | |

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The Harer-Zagier recursion and Ledoux's argument | |

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Quick introduction to Fredholm determinants | |

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The setting, fundamental estimates and definition of the Fredholm determinant | |

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Definition of the Fredholm adjugant, Fredholm resolvent and a fundamental identity | |

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Gap probabilities at 0 and proof of Theorem 3.1.1 | |

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The method of Laplace | |

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Evaluation of the scaling limit: proof of Lemma 3.5.1 | |

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A complement: determinantal relations | |

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Analysis of the sine-kernel | |

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General differentiation formulas | |

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Derivation of the differential equations: proof of Theorem 3.6.1 | |

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Reduction to Painlevï¿½ V | |

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Edge-scaling: proof of Theorem 3.1.4 | |

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Vague convergence of the largest eigenvalue: proof of Theorem 3.1.4 | |

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Steepest descent: proof of Lemma 3.7.2 | |

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Properties of the Airy functions and proof of Lemma 3.7.1 | |

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Analysis of the Tracy-Widom distribution and proof of Theorem 3.1.5 | |

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The first standard moves of the game | |

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The wrinkle in the carpet | |

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Linkage to Painlevï¿½ II | |

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Limiting behavior of the GOE and the GSE | |

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Pfaffians and gap probabilities | |

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Fredholm representation of gap probabilities | |

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Limit calculations | |

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Differential equations | |

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Bibliographical notes | |

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Some generalities | |

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Joint distribution of eigenvalues in the classical matrix ensembles | |

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Integration formulas for classical ensembles | |

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Manifolds, volume measures and the coarea formula | |

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An integration formula of Weyl type | |

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Applications of Weyl's formula | |

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Determinantal point processes | |

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Point processes: basic definitions | |

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Determinantal processes | |

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Determinantal projections | |

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The CLT for determinantal processes | |

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Determinantal processes associated with eigenvalues | |

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Translation invariant determinantal processes | |

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One-dimensional translation invariant determinantal processes | |

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Convergence issues | |

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Examples | |

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Stochastic analysis for random matrices | |

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Dyson's Brownian motion | |

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A dynamical version of Wigner's Theorem | |

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Dynamical central limit theorems | |

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Large deviation bounds | |

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Concentration of measure and random matrices | |

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Concentration inequalities for Hermitian matrices with independent entries | |

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Concentration inequalities for matrices with dependent entries | |

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Tridiagonal matrix models and the ï¿½ ensembles | |

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Tridiagonal representation of ï¿½ ensembles | |

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Scaling limits at the edge of the spectrum | |

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Bibliographical notes | |

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Free probability | |

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Introduction and main results | |

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Noncommutative laws and noncommutative probability spaces | |

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Algebraic noncommutative probability spaces and laws | |

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C*-probability spaces and the weak*-topology | |

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W*-probability spaces | |

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Free independence | |

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Independence and free independence | |

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Free independence and combinatorics | |

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Consequence of free independence: free convolution | |

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Free central limit theorem | |

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Freeness for unbounded variables | |

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Link with random matrices | |

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Convergence of the operator norm of polynomials of independent GUE matrices | |

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Bibliographical notes | |

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Appendices | |

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Linear algebra preliminaries | |

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Identities and bounds | |

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Perturbations for normal and Hermitian matrices | |

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Noncommutative matrix L<sup>p</sup>-norms | |

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Brief review of resultants and discriminants | |

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Topological preliminaries | |

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Generalities | |

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Topological vector spaces and weak topologies | |

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Banach and Polish spaces | |

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Some elements of analysis | |

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Probability measures on Polish spaces | |

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Generalities | |

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Weak topology | |

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Basic notions of large deviations | |

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The skew field H of quaternions and matrix theory over F | |

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Matrix terminology over F and factorization theorems | |

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The spectral theorem and key corollaries | |

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A specialized result on projectors | |

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Algebra for curvature computations | |

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Manifolds | |

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Manifolds embedded in Euclidean space | |

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Proof of the coarea formula | |

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Metrics, connections, curvature, Hessians, and the Laplace-Beltrami operator | |

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Appendix on operator algebras | |

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Basic definitions | |

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Spectral properties | |

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States and positivity | |

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von Neumann algebras | |

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Noncommutative functional calculus | |

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Stochastic calculus notions | |

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References | |

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General conventions and notation | |

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Index | |