This talk establishes 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. More precisely, if the EED of the whole matrix converges to some deterministic probability measure 𝔪, then its difference from the EED of its 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. \\ \\ This is a joint work with Katsunori Fujie.