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

Random matrix theory has led to the discovery of novel matrix models and novel statistical distributions, which are defined by means of Fredholm determinants and which, in many cases, satisfy nonlinear ordinary or partial differential equations. A crucial observation is that these matrix integrals, upon appropriate deformation by means of exponentials containing one or several series of time parameters, satisfy (i) integrable equations and (ii) Virasoro constraints with respect to these time parameters. Most of the time, such matrix integrals can be written — by expressing the integrand in “polar coordinates” — as a multiple integral, which then can be expressed in terms of the determinant of