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.