We discuss an extension of the Jimbo-Miwa-Ueno differential 1-form to a form closed on the full space of extended monodromy data of systems of linear ordinary differential equations with rational coefficients. This extension is based on the results of M. Bertola generalizing a previous construction by B. Malgrange. We show how this 1-form can be used to solve a long-standing problem of evaluation of the connection formulae for the isomonodromic tau functions which would include an explicit computation of the relevant constant factors. We explain how this scheme works for Fuchsian systems and, in particular, calculate the connection constant for generic Painlevé VI tau function. The result proves the conjectural formula for this constant proposed in [ILT13]. We also apply the method to non-Fuchsian systems and evaluate constant factors in the asymptotics of Painlevé II tau function.

Following the work of Conrey, Rubinstein and Snaith [11] and Forrester and Witte [16] we examine a mixed moment of the characteristic polynomial and its derivative for matrices from the unitary group U (N ) (also known as the CUE) and relate the moment to the solution of a Painlevé differential equation. We also calculate a simple form for the asymptotic behaviour of moments of logarithmic derivatives of these characteristic polynomials evaluated near the unit circle.DIMA -DIPARTIMENTO DI MATEMATICA VIA DODECANESO,

Several distribution functions in the classical unitarily invariant matrix ensembles are prime examples of isomonodromic tau functions as introduced by Jimbo, Miwa and Ueno (JMU) in the early 1980s [51]. Recent advances in the theory of tau functions [47], based on earlier works of B. Malgrange and M.Bertola, have allowed to extend the original Jimbo-Miwa-Ueno differential form to a 1-form closed on the full space of extended monodromy data of the underlying Lax pairs. This in turn has yielded a novel approach for the asymptotic evaluation of isomonodromic tau functions, including the exact computation of all relevant constant factors. We use this method to efficiently compute the tail asymptotics of soft-edge, hard-edge and bulk scaled distribution and gap functions in the complex Wishart ensemble, provided each eigenvalue particle has been removed independently with probability 1 − γ ∈ (0, 1].