The eigenvectors of Gaussian matrices with an external source

22 December 2014

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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 (zXt)−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 ⟨ψ0jtj. 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.

Authors

Romain Allez , Joël Bun , Jean-Philippe Bouchaud

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