Moment Matrices and Multi-Component KP, with Applications to Random Matrix Theory

Questions on random matrices and non-intersecting Brownian motions have led to the study of moment matrices with regard to several weights. The main result 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. The main result is a vast generalization of a known fact about infinitesimal deformations of orthogonal polynomials: it concerns an identity between the orthogonality of polynomials on the real line, the bilinear identity in KP theory and a generating functional for the full KP theory. An additional fact not discussed in this paper is that these tau-functions satisfy Virasoro constraints with respect to these time parameters. As one of the many examples worked out in this paper, we consider N non-intersecting Brownian motions in R leaving from the origin, with n(i) particles forced to reach p distinct target points b(i) at time t = 1; of course, Sigma(p)(i=1) n(i) = N. We give a PDE, in terms of the boundary points of the interval E, for the probability that the Brownian particles all pass through an interval E at time 0 < t < 1. It is given by the determinant of a (p + 1) x (p + 1) matrix, which is nearly a wronskian. This theory is also applied to biorthogonal polynomials and orthogonal polynomials on the circle.