Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 083, 20 pages      arXiv:1104.0153      https://doi.org/10.3842/SIGMA.2016.083
Contribution to the Special Issue on Asymptotics and Universality in Random Matrices, Random Growth Processes, Integrable Systems and Statistical Physics in honor of Percy Deift and Craig Tracy

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

Folkmar Bornemann
Zentrum Mathematik - M3, Technische Universität München, 80290 München, Germany

Received April 15, 2016, in final form August 16, 2016; Published online August 19, 2016

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).

Key words: determinantal point processes; Sturm-Liouville operators; scaling limits; strong operator convergence; classical random matrix ensembles; GUE; LUE; JUE; MANOVA.

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