Moment matrices and multi-component KP, with applications to random matrix theory

Abstract

Questions on random matrices and on non-intersecting Brownian motions have led to the study of moment matrices with regard to several weights. The purpose of this paper is to show that the determinants of such moment matrices satisfy, upon adding one set of time deformations for each weight, the multi-component KP-hierarchy: these determinants are thus "tau-functions" for these integrable hierarchies. The tau-functions, so obtained, with appropriate shifts of the time-parameters (forward and backwards) will be expressed in terms of multiple orthogonal polynomials for these weights and their Cauchy transforms. As an application, the multi-component KP-hierarchy leads to a large set of non-linear PDE's, which are useful in finding partial differential equations for the transition probabilities of certain infinite-dimensional diffusions.