The eigenvectors of Gaussian matrices with an external source

We consider a diffusive matrix process $(X_t)_{t\ge 0}$ defined as $X_t:=A+H_t$ where $A$ is a given deterministic Hermitian matrix and $(H_t)_{t\ge 0}$ is a Hermitian Brownian motion. The matrix $A$ is the "external source" that one would like to estimate from the noisy observation $X_t$ at some time $t>0$. We investigate the relationship between the non-perturbed eigenvectors of the matrix $A$ and the perturbed eigenstates at some time $t$ for the three relevant scaling relations between the time $t$ and the dimension $N$ of the matrix $X_t$. We determine the asymptotic (mean-squared) projections of any given non-perturbed eigenvector $|\psi_j^0\rangle$, associated to an eigenvalue $a_j$ of $A$ which may lie inside the bulk of the spectrum or be isolated (spike) from the other eigenvalues, on the orthonormal basis of the perturbed eigenvectors $|\psi_i^t\rangle,i\neq j$. We derive a Burgers type evolution equation for the local resolvent $(z-X_t)_{ii}^{-1}$, describing the evolution of the local density of a given initial state $|\psi_j ^0\rangle$. We are able to solve this equation explicitly in the large $N$ limit, for any initial matrix $A$. In the case of one isolated eigenvector $|\psi_j^0\rangle$, we prove a central limit Theorem for the overlap $\langle \psi_j^0|\psi_j^t\rangle$. When properly centered and rescaled by a factor $\sqrt{N}$, this overlap converges in law towards a centered Gaussian distribution with an explicit variance depending on $t$. Our method is based on analyzing the eigenvector flow under the Dyson Brownian motion.