i will consider random matrices in the general linear group gl(n;c) distributed according to a heat kernel measure. this may also be described as the distribution of brownian motion in gl(n;c) starting at the identity. numerically, the eigenvalues appear to cluster into a certain domain $\sigma_t$ as $n$ tends to infinity. a natural candidate for the limiting eigenvalue distribution is the “brown measure” of the limiting object, which is biane’s ``free multiplicative brownian motion.'' i will describe recent work with driver and kemp in which we compute this brown measure. the talk will be self contained and will have lots of pictures.