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January 2009 From random matrices to random analytic functions
Manjunath Krishnapur
Ann. Probab. 37(1): 314-346 (January 2009). DOI: 10.1214/08-AOP404


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).


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Manjunath Krishnapur. "From random matrices to random analytic functions." Ann. Probab. 37 (1) 314 - 346, January 2009.


Published: January 2009
First available in Project Euclid: 17 February 2009

zbMATH: 1221.30007
MathSciNet: MR2489167
Digital Object Identifier: 10.1214/08-AOP404

Primary: 15A52 , 30B20
Secondary: 60H25

Keywords: Determinantal process , Haar unitary , hyperbolic plane , invariant point process , Random analytic function , Random matrix , Zeros

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.37 • No. 1 • January 2009
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