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On the Scaling Limits of Determinantal Point Processes with Kernels Induced by Sturm-Liouville Operators
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2016, том 12
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On the Scaling Limits of Determinantal Point Processes with Kernels Induced by Sturm-Liouville Operators
Інший ID:
2010 Mathematics Subject Classification: 15B52; 34B24; 33C45
DOI:10.3842/SIGMA.2016.083
DOI:10.3842/SIGMA.2016.083
Посилання:
On the Scaling Limits of Determinantal Point Processes with Kernels Induced by Sturm-Liouville Operators / F. Bornemann // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 25 назв. — англ.
Підтримка:
This paper is a 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.
The full collection is available at http://www.emis.de/journals/SIGMA/Deift-Tracy.html.
Дата:
2016
Переглядів:
493
Завантажень:
246
Анотація:
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).
