ASYMPTOTIC DISTRIBUTION OF COMPLEX ZEROS OF RANDOM ANALYTIC FUNCTIONS

• 成果类型: Article
• 署名作者: Kabluchko, Zakhar; Zaporozhets, Dmitry
• 署名单位: Ulm University; Russian Academy of Sciences; Steklov Mathematical Institute of the Russian Academy of Sciences; St. Petersburg Department of the Steklov Mathematical Institute of the Russian Academy of Sciences
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/13-AOP847
• 发表日期: 2014
• 页码: 1374-1395
• 关键词: random polynomials; roots; UNIVERSALITY

摘要

Let xi(0), xi(1,) . . . be independent identically distributed complex-valued random variables such that E log(1 + vertical bar xi(0)vertical bar) < infinity. We consider random analytic functions of the form G(n)(z) = Sigma xi(k)f(k,n)Z(k), where f(k,n) are deterministic complex coefficients. Let mu(n) be the random measure counting the complex zeros of G(n) according to their multiplicities. Assuming essentially that -1/nlog f([tn]),(n) -> u(t) as n -> infinity, where u(t) is some function, we show that the measure 1/n mu(n).An converges in probability to some deterministic measure mu which is characterized in terms of the Legendre Fenchel transform of u. The limiting measure mu does not depend on the distribution of the xi(k)'s. This result is applied to several ensembles of random analytic functions including the ensembles corresponding to the three two-dimensional geometries of constant curvature. As another application, we prove a random polynomial analogue of the circular law for random matrices.