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 $\mathfrak{m}$, 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 $\mathfrak{m}$ 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 $\mathfrak{m}$. This formula is more or less known, but we provide a different proof by observing a combinatorial structure of non-crossing partitions.