  In this paper, we study random matrix models which are obtained as a
non-commutative polynomial in random matrix variables of two kinds: (a) a first
kind which have a discrete spectrum in the limit, (b) a second kind which have
a joint limiting distribution in Voiculescu's sense and are globally
rotationally invariant. We assume that each monomial constituting this
polynomial contains at least one variable of type (a), and show that this
random matrix model has a set of eigenvalues that almost surely converges to a
deterministic set of numbers that is either finite or accumulating to only zero
in the large dimension limit. For this purpose we define a framework (cyclic
monotone independence) for analyzing discrete spectra and develop the moment
method for the eigenvalues of compact (and in particular Schatten class)
operators. We give several explicit calculations of discrete eigenvalues of our
model.
