Statistical Mechanics and Random Matrices

نویسنده

  • Alice Guionnet
چکیده

Statistical Mechanics and Random Matrices 3 1. Introduction 6 2. Motivations 7 3. The different scales; typical results 12 Lecture 1. Wigner matrices and moments estimates 15 1. Wigner's theorem 16 2. Words in several independent Wigner matrices 23 3. Estimates on the largest eigenvalue of Wigner matrices 25 Lecture 2. Gaussian Wigner matrices and Fredholm determinants 27 1. Joint law of the eigenvalues 28 2. Joint law of the eigenvalues and determinantal law 29 3. Determinantal structure and Fredholm determinants 31 4. Fredholm determinant and asymptotics 31 Lecture 3. Wigner matrices and concentration inequalities 35 1. Concentration inequalities and logarithmic Sobolev inequalities 36 2. Smoothness and convexity of the eigenvalues of a matrix and of traces of matrices 39 3. Concentration inequalities for random matrices 42 4. Brascamp-Lieb inequalities; Applications to random matrices 43 Lecture 4. Matrix models 49 1. Combinatorics of maps and non-commutative polynomials 51 2. Formal expansion of matrix integrals 55 3. First order expansion for the free energy 59 4. Discussion 66 Lecture 5. Random matrices and dynamics 69 1. Free Brownian motions and related stochastic differential calculus 70 2. Consequences 76 3. Discussion 78 Bibliography 81 6 LECTURE 0. CONTENTS

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تاریخ انتشار 2008