Asymptotic distribution of complex zeros of random analytic functions

Abstract

Let ${\xi }_0,{\xi }_1,\dots$ be independent identically distributed complex- valued random variables such that $\mathbb{E}\log (1+|{\xi }_0|)<\infty$. We consider random analytic functions of the form \[\mathbf{G}_n(z)=\sum_{k=0}^{\infty}ξ_kf_{k,n}z^k,\] where $f_{k,n}$ are deterministic complex coefficients. Let ${\mu }_n$ be the random measure counting the complex zeros of ${\mathbf{G}}_n$ according to their multiplicities. Assuming essentially that $-\frac{1}{n}\log f_{[tn],n}\to u(t)$ as $n\to \infty$, where $u(t)$ is some function, we show that the measure $\frac{1}{n}{\mu }_n$ 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.

Published in at http://dx.doi.org/10.1214/13-AOP847 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org). arXiv admin note: substantial text overlap with arXiv:1205.5355

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• Real zeros of random analytic functions associated with geometries of constant curvature
• The Common Limit of the Linear Statistics of Zeros of Random Polynomials and Their Derivatives
• Random Polynomials in Several Complex Variables
• Universality of the minimum modulus for random trigonometric polynomials
• Zeros of random polynomials undergoing the heat flow
• Geometric quantization results for semi-positive line bundles on a Riemann surface
• The "pits effect" for entire functions of exponential type and the Wiener spectrum
• Zero distribution of power series and binary correlation of coefficients
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• Zeros of repeated derivatives of random polynomials
• The limit empirical spectral distribution of complex matrix polynomials
• Distances between zeroes and critical points for random polynomials with i.i.d. zeroes
• Zero-free neighborhood around the unit circle for Kac polynomials
• Correlations between real and complex zeros of a random polynomial