Moment determinants as isomonodromic tau functions

We consider a wide class of determinants whose entries are moments of the so-called semiclassical functionals and we show that they are tau functions for an appropriate isomonodromic family which depends on the parameters of the symbols for the functionals.This shows that the vanishing of the tau function for those systems is the obstruction to the solvability of a Riemann-Hilbert problem associated with certain classes of (multiple) orthogonal polynomials.The determinants include Hankel, Toeplitz and shifted-Toeplitz determinants as well as determinants of bimoment functionals and the determinants arising in the study of multiple orthogonality.Some of these determinants also appear as partition functions of random matrix models, including an instance of a two-matrix model.