The eigenvectors of Gaussian matrices with an external source
We consider a diffusive matrix process (Xt)t≥0 defined as Xt:=A+Ht where A is a given deterministic Hermitian matrix and (Ht)t≥0 is a Hermitian Brownian motion. The matrix A is the 'external source' that one would like to estimate from the noisy observation Xt 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 Xt. We determine the asymptotic (mean-squared) projections of any given non-perturbed eigenvector |ψ0j⟩, associated to an eigenvalue aj 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 |ψti⟩,i≠j. We derive a Burgers type evolution equation for the local resolvent (z−Xt)−1ii, describing the evolution of the local density of a given initial state |ψ0j⟩. 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 |ψ0j⟩, we prove a central limit Theorem for the overlap ⟨ψ0j|ψtj⟩. When properly centered and rescaled by a factor 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.