Fermionic observable for the eigenvector moment flow and fluctuations of eigenvectors of random matrices 

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Analysis of the perfect matching observable

In this subsection we will again condition on the event A of good eigenvalue paths where the local laws and the finite speed of propagation holds. Consider now a deterministic set of indices I [[1;N]].
Note that in the definition of the centered overlaps pii, we can only center by a constant not depending on i. However in Theorem 2.1.7, one can see that the expectation of the probability mass of the itheigenvector on I clearly depends on i. Thus, we will need to localize our perfect matching observables onto a window of size w chosen later and show that these pii are, up to an error depending on w, centered around the same constant. The size of the window w will be taken so that N w Nt similarly to the previous section. More precisely, we will fix an integer i0 2 Ar and consider the set of indices A w(i0) = fi 2 [[1;N]]; t;i 2 [ t;i0 􀀀 (1 􀀀 )w; t;i0 + (1 􀀀 )w]g.

Continuity of the Dyson Brownian motion

In Subsection 2.3.1 we showed that the moments of the eigenvectors of the matrix H are asymptotically those of a Gaussian random variable with variance 2 t . If we would have taken the time t􀀀 from the start, the previous section gives us (2.3.15) for W a matrix from the Gaussian Orthogonal Ensemble.
Now, since is a small time, recall that t, we can use the continuity of the Dyson Brownian motion to show that H and H0 = Wt have the same local statistics. In order to state a proper continuity lemma we need to have a dynamics with constant second moments and vanishing expectation. First see that the variance of the centered model is E h (Wt;ij 􀀀 Dij)2 i = t N .

Reverse heat flow

In Subsection 2.3.2, we showed Theorem 2.3.8, which corresponds to our main result for the matrix H = Wt + pGOE for N􀀀1 t with a general Wigner matrix W in the definition of Wt. Thus, the overwhelming probability bound holds for the eigenvectors of this matrix H giving us a strong form of quantum unique ergodicity for the deformed Gaussian divisible ensemble. In order to remove the small Gaussian component in the matrix, we will use the reverse heat flow technique from [EPR+10, ESY11] which allows us to obtain an error as small as we want in total variation between two matrix ensembles. In order to use this technique, we need the smoothness assumption on the matrix W given by Definition 2.1.6. We first introduce some notation for this section.
As before, we let denote the distribution of the entries of W, and ‘ denote the density of with respect to , the Gaussian distribution with mean zero and variance one, that is, d = ‘d%. The reverse heat flow technique gives the existence of a probability distribution ~s for any s small enough such that making ~s undergo the Ornstein-Uhlenbeck process of generator A := 1 2 @2 @x2 􀀀 x 2 @ @x.

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Construction of the Fermionic observable

We will be able to construct an observable based on the families of Grassman variables which will follow (3.1.18). Then by taking the Gaussian expectation defined in (3.2.2) we will obtain the observable (3.1.17) by choosing the right covariance matrix . In the following definitions we will fix a set of indices I [[1;N]] and consider (qi)i2I a family of vectors of RN not necesarily orthogonal. We will consider the observable, for us the solution to the Dyson vector flow (3.1.14)

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 

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