The Annals of Probability

From random matrices to random analytic functions

Manjunath Krishnapur

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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

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

First available in Project Euclid: 17 February 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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

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