The Annals of Probability
- Ann. Probab.
- Volume 37, Number 1 (2009), 314-346.
From random matrices to random analytic functions
We consider two families of random matrix-valued analytic functions: (1) G1−zG2 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).
Ann. Probab., Volume 37, Number 1 (2009), 314-346.
First available in Project Euclid: 17 February 2009
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 30B20: Random power series 15A52
Secondary: 60H25: Random operators and equations [See also 47B80]
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