  In this paper two independent and unitarily invariant projection matrices
P(N) and Q(N) are considered and the large deviation is proven for the
eigenvalue density of all polynomials of them as the matrix size $N$ converges
to infinity. The result is formulated on the tracial state space $TS({\cal A})$
of the universal $C^*$-algebra ${\cal A}$ generated by two selfadjoint
projections. The random pair $(P(N),Q(N))$ determines a random tracial state
$\tau_N \in TS({\cal A})$ and $\tau_N$ satisfies the large deviation. The rate
function is in close connection with Voiculescu's free entropy defined for
pairs of projections.
