A First Course in Random Matrix Theory

Copyright
Contents
Preface
List of Symbols
Part I Classical Random Matrix Theory
	1 Deterministic Matrices
		1.1 Matrices, Eigenvalues and Singular Values
		1.2 Some Useful Theorems and Identities
	2 Wigner Ensemble and Semi-Circle Law
		2.1 Normalized Trace and Sample Averages
		2.2 The Wigner Ensemble
		2.3 Resolvent and Stieltjes Transform
	3 More on Gaussian Matrices*
		3.1 Other Gaussian Ensembles
		3.2 Moments and Non-Crossing Pair Partitions
	4 Wishart Ensemble and Marˇ cenko–Pastur Distribution
		4.1 Wishart Matrices
		4.2 Marˇ cenko–Pastur Using the Cavity Method
	5 Joint Distribution of Eigenvalues
		5.1 From Matrix Elements to Eigenvalues
		5.2 Coulomb Gas and Maximum Likelihood Configurations
		5.3 Applications: Wigner, Wishart and the One-Cut Assumption
		5.4 Fluctuations Around the Most Likely Configuration
		5.5 An Eigenvalue Density Saddle Point
	6 Eigenvalues and Orthogonal Polynomials*
		6.1 Wigner Matrices and Hermite Polynomials
		6.2 Laguerre Polynomials
		6.3 Unitary Ensembles
	7 The Jacobi Ensemble*
		7.1 Properties of Jacobi Matrices
		7.2 Jacobi Matrices and Jacobi Polynomials
Part II Sums and Products of Random Matrices
	8 Addition of Random Variables and Brownian Motion
		8.1 Sums of Random Variables
		8.2 Stochastic Calculus
	9 Dyson Brownian Motion
		9.1 Dyson Brownian Motion I: Perturbation Theory
		9.2 Dyson Brownian Motion II: Itoˆ Calculus
		9.3 The Dyson Brownian Motion for the Resolvent
		9.4 The Dyson Brownian Motion with a Potential
		9.5 Non-Intersecting Brownian Motions and the Karlin–McGregor Formula
	10 Addition of Large Random Matrices
		10.1 Adding a Large Wigner Matrix to an Arbitrary Matrix
		10.2 Generalization to Non-Wigner Matrices
		10.3 The Rank-1 HCIZ Integral
		10.4 Invertibility of the Stieltjes Transform
		10.5 The Full-Rank HCIZ Integral
	11 Free Probabilities
		11.1 Algebraic Probabilities: Some Definitions
		11.2 Addition of Commuting Variables
		11.3 Non-Commuting Variables
		11.4 Free Product
	12 Free Random Matrices
		12.1 Random Rotations and Freeness
		12.2 R-Transforms and Resummed Perturbation Theory
		12.3 The Central Limit Theorem for Matrices
		12.4 Finite Free Convolutions
		12.5 Freeness for 2 × 2 Matrices
	13 The Replica Method*
		13.1 Stieltjes Transform
		13.2 Resolvent Matrix
		13.3 Rank-1 HCIZ and Replicas
		13.4 Spin-Glasses, Replicas and Low-Rank HCIZ
	14 Edge Eigenvalues and Outliers
		14.1 The Tracy–Widom Regime
		14.2 Additive Low-Rank Perturbations
		14.3 Fat Tails
		14.4 Multiplicative Perturbation
		14.5 Phase Retrieval and Outliers
Part III Applications
15 Addition and Multiplication: Recipes and Examples
	15.1 Summary
	15.2 R- and S-Transforms and Moments of Useful Ensembles
	15.3 Worked-Out Examples: Addition
	15.4 Worked-Out Examples: Multiplication
16 Products of Many Random Matrices
	16.1 Products of Many Free Matrices
	16.2 The Free Log-Normal
	16.3 A Multiplicative Dyson Brownian Motion
	16.4 The Matrix Kesten Problem
17 Sample Covariance Matrices
	17.1 Spatial Correlations
	17.2 Temporal Correlations
	17.3 Time Dependent Variance
	17.4 Empirical Cross-Covariance Matrices
18 Bayesian Estimation
	18.1 Bayesian Estimation
	18.2 Estimating a Vector: Ridge and LASSO
	18.3 Bayesian Estimation of the True Covariance Matrix
19 Eigenvector Overlaps and Rotationally Invariant Estimators
	19.1 Eigenvector Overlaps
	19.2 Rotationally Invariant Estimators
	19.3 Properties of the Optimal RIE for Covariance Matrices
	19.4 Conditional Average in Free Probability
	19.5 Real Data
	19.6 Validation and RIE
20 Applications to Finance
	20.1 Portfolio Theory
	20.2 The High-Dimensional Limit
	20.3 The Statistics of Price Changes: A Short Overview
	20.4 Empirical Covariance Matrices
Appendix Mathematical Tools
	A.1 Saddle Point Method
	A.2 Tricomi’s Formula
	A.3 Toeplitz and Circulant Matrices
Index