The Annals of Probability

From random matrices to random analytic functions

Manjunath Krishnapur

Full-text: Open access

Abstract

We consider two families of random matrix-valued analytic functions: (1) G1zG2 and (2) G0+zG1+z2G2+⋯, where Gi are n×n random matrices with independent standard complex Gaussian entries. The random set of z where these matrix-analytic functions become singular is shown to be determinantal point processes in the sphere and the hyperbolic plane, respectively. The kernels of these determinantal processes are reproducing kernels of certain Hilbert spaces (“Bargmann–Fock spaces”) of holomorphic functions on the corresponding surfaces. Along with the new results, this also gives a unified framework in which to view a theorem of Peres and Virág (n=1 in the second setting above) and a well-known result of Ginibre on Gaussian random matrices (which may be viewed as an analogue of our results in the whole plane).

Article information

Source
Ann. Probab., Volume 37, Number 1 (2009), 314-346.

Dates
First available in Project Euclid: 17 February 2009

Permanent link to this document
https://projecteuclid.org/euclid.aop/1234881692

Digital Object Identifier
doi:10.1214/08-AOP404

Mathematical Reviews number (MathSciNet)
MR2489167

Zentralblatt MATH identifier
1221.30007

Subjects
Primary: 30B20: Random power series 15A52
Secondary: 60H25: Random operators and equations [See also 47B80]

Keywords
Random analytic function zeros determinantal process random matrix Haar unitary hyperbolic plane invariant point process

Citation

Krishnapur, Manjunath. From random matrices to random analytic functions. Ann. Probab. 37 (2009), no. 1, 314--346. doi:10.1214/08-AOP404. https://projecteuclid.org/euclid.aop/1234881692


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