  We study multiplicative statistics for the eigenvalues of unitarily-invariant
Hermitian random matrix models. We consider one-cut regular polynomial
potentials and a large class of multiplicative statistics. We show that in the
large matrix limit several associated quantities converge to limits which are
universal in both the potential and the family of multiplicative statistics
considered. In turn, such universal limits are described by the
integro-differential Painlev\'e II equation, and in particular they connect the
random matrix models considered with the narrow wedge solution to the KPZ
equation at any finite time.
