Darboux Transformations and Random Point Processes

In this paper, we describe a general method to derive formulas relating the gap probabilities of some classical determinantal random point processes (Airy, Pearcey, and Hermite) with the gap probability of the same processes with "wanderers", "inliers", and "outliers". In this way, we generalize the Painleve-like formula found by Baik for the Baik-Ben Arous-Peche distribution to many different cases, both in the one and multi-time setting. The method is not ad hoc and relies upon the notion of Schlesinger transformations for Riemann-Hilbert problems.