On the analysis of incomplete spectra in random matrix theory through an extension of the Jimbo–Miwa–Ueno differential

Several distribution functions in the classical unitarily invari-ant matrix ensembles are prime examples of isomonodromic tau functions as introduced by Jimbo, Miwa and Ueno (JMU) in the early 1980s [45]. Recent advances in the theory of tau functions [41], 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, in-cluding the exact computation of all relevant constant factors. We use this method to efficiently compute the tail asymp-totics of soft-edge, hard-edge and bulk scaled distribution and gap functions in the complex Wishart ensemble, provided