Differential Equations for Dyson Processes

Abstract

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×n Hermitian matrices, and the eigenvalues describe n curves. Given sets X₁,...,X _m the probability that for each k no curve passes through X _k at time τ _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 Prähofer 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 ξ whose solution determined the probabilities

where τ→A(τ) 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 τ _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.