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

TL;DR

This paper investigates the scaling limits of determinantal point processes derived from Sturm-Liouville operators, establishing universality of Dyson, Airy, and Bessel kernels across different spectral regimes and providing unified derivations for classical random matrix ensembles.

Contribution

It introduces a new approach focusing on strong operator convergence of spectral projection kernels, unifying the derivation of scaling limits for various random matrix models.

Findings

Dyson, Airy, and Bessel kernels are universal in their respective scaling limits.

The method provides a unified derivation of classical random matrix ensemble limits.

Strong convergence of integral operators replaces kernel function convergence as the key concept.

Abstract

By applying an idea of Borodin and Olshanski [J. Algebra 313 (2007), 40-60], 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 random matrix ensembles with unitary invariance, that is, the Gaussian unitary ensemble (GUE), the Wishart or Laguerre unitary ensemble (LUE), and the MANOVA (multivariate analysis of variance) or Jacobi unitary ensemble (JUE).