SU(1, 1) Random Polynomials
 Pavel Bleher,
 Denis Ridzal
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Abstract
We study statistical properties of zeros of random polynomials and random analytic functions associated with the pseudoeuclidean group of symmetries SU(1, 1), by utilizing both analytical and numerical techniques. We first show that zeros of the SU(1, 1) random polynomial of degree N are concentrated in a narrow annulus of the order of N ^{−1} around the unit circle on the complex plane, and we find an explicit formula for the scaled density of the zeros distribution along the radius in the limit N→∞. Our results are supported through various numerical simulations. We then extend results of Hannay^{(1)} and Bleher et al. ^{(2)} to derive different formulae for correlations between zeros of the SU(1, 1) random analytic functions, by applying the generalized Kac–Rice formula. We express the correlation functions in terms of some Gaussian integrals, which can be evaluated combinatorially as a finite sum over Feynman diagrams or as a supersymmetric integral. Due to the SU(1, 1) symmetry, the correlation functions depend only on the hyperbolic distances between the points on the unit disk, and we obtain an explicit formula for the two point correlation function. It displays quadratic repulsion at small distances and fast decay of correlations at infinity. In an appendix to the paper we evaluate correlations between the outer zeros z _{ j }>1 of the SU(1, 1) random polynomial, and we prove that the inner and outer zeros are independent in the limit when the degree of the polynomial goes to infinity.
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 Title
 SU(1, 1) Random Polynomials
 Journal

Journal of Statistical Physics
Volume 106, Issue 12 , pp 147171
 Cover Date
 20020101
 DOI
 10.1023/A:1013124213145
 Print ISSN
 00224715
 Online ISSN
 15729613
 Publisher
 Kluwer Academic PublishersPlenum Publishers
 Additional Links
 Topics
 Keywords

 random polynomial
 pseudosphere
 correlations between zeros
 Industry Sectors
 Authors

 Pavel Bleher ^{(1)}
 Denis Ridzal ^{(1)}
 Author Affiliations

 1. Department of Mathematical Sciences, Indiana University–Purdue University Indianapolis, 402 N. Blackford Street, Indianapolis, Indiana, 462023216