Interlacing eigenvectors of large Gaussian matrices
Elie Attal, Romain Allez · Journal of Physics A Mathematical and Theoretical · 2024
Abstract We consider the eigenvectors of the principal minor of dimension n < N of the Dyson Brownian motion in R N and investigate their asymptotic overlaps with the eigenvectors of the full matrix in the limit of large dimension. We explicitly compute the limiting rescaled mean squared overlaps in the large n , N limit with n / N tending to a fixed ratio q, for any initial symmetric matrix A. This is accomplished using a Burgers-type evolution equation for a specific resolvent. In the Gaussian orthogonal ensemble case, our formula simplifies, and we identify an eigenvector analogue of the well-known interlacing of eigenvalues. We investigate in particular the case where A has isolated eigenvalues. Our method is based on analysing the eigenvector flow under the Dyson Brownian motion.