Asymptotics of Random Matrices and Related Models

Description

Probability theory is based on the notion of independence. The celebrated law of large numbers and the central limit theorem describe the asymptotics of the sum of independent variables. However, there are many models of strongly correlated random variables: for instance, the eigenvalues of random matrices or the tiles in random tilings. Classical tools of probability theory are useless to study such models. These lecture notes describe a general strategy to study the fluctuations of strongly interacting random variables. This strategy is based on the asymptotic analysis of Dyson-Schwinger (or loop) equations: the author will show how these equations are derived, how to obtain the concentration of measure estimates required to study these equations asymptotically, and how to deduce from this analysis the global fluctuations of the model. The author will apply this strategy in different settings: eigenvalues of random matrices, matrix models with one or several cuts, random tilings, and several matrices models. Cover Title page Preface Chapter 1. Introduction Chapter 2. The example of the GUE Chapter 3. Wigner random matrices 3.1. Law of large numbers: Light tails 3.2. Law of large numbers: Heavy tails 3.3. CLT Chapter 4. Beta-ensembles 4.1. Law of large numbers and large deviation principles 4.2. Concentration of measure 4.3. The Dyson-Schwinger equations 4.4. Expansion of the partition function 4.5. The Stieltjes transforms approach Chapter 5. Discrete beta-ensembles 5.1. Large deviations, law of large numbers 5.2. Concentration of measure 5.3. Nekrasov’s equations 5.4. Second order expansion of linear statistics 5.5. Expansion of the partition function Chapter 6. Continuous beta-models: The several cut case 6.1. The fixed filling fractions model 6.2. Central limit theorem for the full model Chapter 7. Several matrix-ensembles 7.1. Non-commutative derivatives 7.2. Non-commutative Dyson-Schwinger equations 7.3. Independent GUE matrices 7.4. Several interacting matrices models 7.5. Second order expansion for the free energy Chapter 8. Universality for beta-models Bibliography Index Back Cover