John Harnad - The bilinear differential equations satisfied by Fredholm determinants appearing in random matrices

Bilinear differential equations satisfied by the Fredholm determinants of certain integral operators appearing as spectral distribution functions for random matrices are deduced from the associated systems of nonautonomous Hamiltonian equations satisfied by auxiliary canonical phase space variables introduced by Tracy and Widom. The essential step is to express the latter as isomonodromic deformation equations for families of rational covariant derivative operators on the Riemann sphere and to interpret the Fredholm determinants as isomonodromic -functions.