The Spectra of Principal Submatrices in Rotationally Invariant Hermitian Random Matrices and the Markov– Krein Correspondence

We prove a concentration phenomenon on the empirical eigenvalue distribution (EED) of the principal submatrix in a random hermitian matrix whose distribution is invariant under unitary conjugacy; for example, this class includes GUE (Gaussian Unitary Ensemble) and Wishart matrices. More precisely, if the EED of the whole matrix converges to some deterministic probability measure m, then its fluctuation from the EED of the principal submatrix, after a rescaling, concentrates at the Rayleigh measure (in general, a Schwartz distribution) associated with m by the Markov–Krein correspondence. For the proof, we use the moment method with Weingarten calculus and free probability. At some stage of calculations, the proof requires a relation between the moments of the Rayleigh measure and free cumulants of m. This formula is more or less known, but we provide a different proof by observing a combinatorial structure of non-crossing partitions.