Heat kernel measure $K(t, I, A) dA$ on positive Hermitian $NxN$ matrices is a probability measure whose large $N$ limit is important for several different types of problems in mathematical physics. My talk introduces a new application: to random Kahler metrics on any Kahler manifold. The pair correlation function of random metrics is explicitly calculated for each $N$. The large $N$ asymptotics are closely related to zero sets of random holomorphic functions.