Correlations between zeros of non-Gaussian random polynomials

• Published in: International Mathematics Research Notices 2004, Vol.2004 (46), p.2443-2484
• Main Authors: Bleher, Pavel, Di, Xiaojun
• Format: Article
• Language: English
• ISSN: 1073-7928; 1687-1197; 1687-0247
• DOI: 10.1155/S1073792804132418

The existence of the scaling limit and its universality for correlations between zeros of Gaussian random polynomials, or more generally, Gaussian random sections of powers of a line bundle over a compact manifold has been proved in a great generality in the works of Bogomolny, Bohigas, Lebœuf, Hannay, Bleher, Schiffman, Zelditch, and others. In the present work, we prove the existence of the scaling limit for a class of non-Gaussian random polynomials. Our main result is that away from the origin, the scaling limit exists and is universal so that it does not depend on the distribution of the coefficients. At the origin, the scaling limit is not universal, and we find a crossover from the nonuniversal asymptotics of the density of the probability distribution of zeros at the origin to the universal one away from the origin.