  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.
