  Given a selfadjoint polynomial $P(X,Y)$ in two noncommuting selfadjoint
indeterminates, we investigate the asymptotic eigenvalue behavior of the random
matrix $P(A\_N,B\_N)$, where $A\_N$ and $B\_N$ are independent Hermitian random
matrices and the distribution of $B\_N$ is invariant under conjugation by
unitary operators. We assume that the empirical eigenvalue distributions of
$A\_N$ and $B\_N$ converge almost surely to deterministic probability measures
$\mu $ and $\nu$, respectively. In addition, the eigenvalues of $A\_N$ and
$B\_N$ are assumed to converge uniformly almost surely to the support of $\mu$
and $\nu,$ respectively, except for a fixed finite number of fixed eigenvalues
(spikes) of $A\_N$. It is known that almost surely the empirical distribution
of the eigenvalues of $P(A\_N,B\_N)$ converges to a certain deterministic
probability measure $\eta$ (sometimes denoted $\eta=P^\square(\mu,\nu)$) and,
when there are no spikes, the eigenvalues of $P(A\_N,B\_N)$ converge uniformly
almost surely to the support of $\eta$. When spikes are present, we show that
the eigenvalues of $P(A\_N,B\_N)$ still converge uniformly to the support of
$\eta$, with the possible exception of certain isolated outliers whose location
can be determined in terms of $\mu,\nu,P$, and the spikes of $A\_N$. We
establish a similar result when $B\_N$ is replaced by a Wigner matrix. The
relation between outliers and spikes is described using the operator-valued
subordination functions of free probability theory. These results extend known
facts from the special case in which $P(X,Y)=X+Y$.
