The Annals of Probability

Asymptotic distribution of complex zeros of random analytic functions

Zakhar Kabluchko and Dmitry Zaporozhets

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Abstract

Let $\xi_{0},\xi_{1},\ldots$ 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 \[\mathbf{G}_{n}(z)=\sum_{k=0}^{\infty}\xi_{k}f_{k,n}z^{k},\] where $f_{k,n}$ are deterministic complex coefficients. Let $\mu_{n}$ be the random measure counting the complex zeros of $\mathbf{G}_{n}$ 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.

Article information

Source
Ann. Probab. Volume 42, Number 4 (2014), 1374-1395.

Dates
First available in Project Euclid: 3 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.aop/1404394067

Digital Object Identifier
doi:10.1214/13-AOP847

Mathematical Reviews number (MathSciNet)
MR3262481

Zentralblatt MATH identifier
1295.30008

Subjects
Primary: 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10}
Secondary: 30B20: Random power series 26C10: Polynomials: location of zeros [See also 12D10, 30C15, 65H05] 60G57: Random measures 60B10: Convergence of probability measures 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

Keywords
Random analytic function random polynomial random power series empirical distribution of zeros circular law logarithmic potential equilibrium measure Legendre–Fenchel transform

Citation

Kabluchko, Zakhar; Zaporozhets, Dmitry. Asymptotic distribution of complex zeros of random analytic functions. Ann. Probab. 42 (2014), no. 4, 1374--1395. doi:10.1214/13-AOP847. https://projecteuclid.org/euclid.aop/1404394067


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