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Table of contents
I Dynamics of eigenvectors of random matrices
1 Introduction
1.1 Invariant Ensembles
1.2 Wigner ensembles
1.2.1 Presentation of the model and universality results
1.2.2 Method of proof
1.3 Other mean-field models
1.3.1 General mean and covariance
1.3.2 Adjacency matrices of random graphs
1.3.3 Lévy matrices
1.3.4 Addition of random matrices
1.3.5 Deformed Wigner matrices
2 Eigenvector distribution and quantum unique ergodicity for deformed Wigner matrices
2.1 Introduction
2.1.1 Main Results
2.1.2 Method of Proof
2.2 Local laws
2.2.1 Anisotropic local law for deformed Wigner matrices
2.3 Short time relaxation
2.3.1 Analysis of the moment observable
2.3.2 Analysis of the perfect matching observable
2.4 Approximation by a Gaussian divisible ensemble
2.4.1 Continuity of the Dyson Brownian motion
2.4.2 Reverse heat flow
2.5 Proofs of main results
3 Fermionic observable for the eigenvector moment flow and fluctuations of eigenvectors of random matrices
3.1 Introduction
3.1.1 Main results
3.1.2 Method of proof
3.2 Proof of Theorem 3.1.10
3.2.1 Preliminaries
3.2.2 Construction of the Fermionic observable
3.3 Relaxation by the Dyson Brownian motion
3.4 Proof of Theorem 3.1.4
3.5 Combinatorial proof of Theorem 3.1.10
3.6 Case of Hermitian matrices
II Eigenvalues of nonlinear matrix models
1 Introduction
1.1 Sample covariance matrices
1.1.1 Wishart distribution
1.1.2 Universality results
1.2 The method of moments
2 Eigenvalue distribution of nonlinear matrix models
2.1 Introduction
2.2 Description of the model
2.3 Moment method when f is a polynomial
2.3.1 Case where f is a monomial of odd degree
2.3.2 Case where f is a monomial of even degree
2.3.3 Case where f is a polynomial
2.3.4 Convergence of moments in probability
2.3.5 Passage to sub-Gaussian random variables
2.3.6 Weak convergence of the empirical spectral measure
2.3.7 Recursion relation for the Stieltjes transform
2.4 Polynomial approximation for general activation function
2.5 Behavior of the largest eigenvalue
2.5.1 Convergence of the largest eigenvalue to the edge of the support
2.6 Propagation of eigenvalue distribution through multiple layers
2.6.1 Eigenvalue distribution of Y (2)
2.6.2 Invariance of the distribution in the case when 2(f) vanishes
Bibliography
