Darboux transformations and random point processes

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