Differential equations for Dyson processes

We call a Dyson process any process on ensembles of matrices in which the entries undergo diffusion. We are interested in the distribution of the eigenvalues ( or singular values) of such matrices. In the original Dyson process it was the ensemble of n x n Hermitian matrices, and the eigenvalues describe n curves. Given sets X-1,..., X-m the probability that for each k no curve passes through X-k at time tau(k) is given by the Fredholm determinant of a certain matrix kernel, the extended Hermite kernel. For this reason we call this Dyson process the Hermite process. Similarly, when the entries of a complex matrix undergo diffusion we call the evolution of its singular values the Laguerre process, for which there is a corresponding extended Laguerre kernel. Scaling the Hermite process at the edge leads to the Airy process ( which was introduced by Prahofer and Spohn as the limiting stationary process for a polynuclear growth model) and in the bulk to the sine process; scaling the Laguerre process at the edge leads to the Bessel process. In earlier work the authors found a system of ordinary differential equations with independent variable. whose solution determined the probabilities Pr (A(tau(1)) <ξ(1)+ξ.,..., A(τ(m)) < xi(m) +xi), where tau --> A(tau) denotes the top curve of the Airy process. Our first result is a generalization and strengthening of this. We assume that each X-k is a finite union of intervals and find a system of partial differential equations, with the end-points of the intervals of the X-k as independent variables, whose solution determines the probability that for each k no curve passes through X-k at time tau(k). Then we find the analogous systems for the Hermite process ( which is more complicated) and also for the sine process. Finally we find an analogous system of PDEs for the Bessel process, which is the most difficult.