On the Scaling Limits of Determinantal Point Processes with Kernels Induced by Sturm-Liouville Operators

By applying an idea of Borodin and Olshanski (2007), we study various scaling limits of determinantal point processes with trace class projection kernels given by spectral projections of selfadjoint Sturm-Liouville operators. Instead of studying the convergence of the kernels as functions, the method directly addresses the strong convergence of the induced integral operators. We show that, for this notion of convergence, the Dyson, Airy, and Bessel kernels are universal in the bulk, soft-edge, and hard-edge scaling limits. This result allows us to give a short and unified derivation of the known formulae for the scaling limits of the classical unitary random matrix ensembles (GUE, LUE/Wishart, JUE/MANOVA).