Asymptotic distribution of complex zeros of random analytic functions
By Zakhar Kabluchko, Dmitry Zaporozhets. 2014
91 citations recorded by the source corpus.
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 \[\mathbf{G}_n(z)=\sum_{k=0}^{\infty}\xi_kf_{k,n}z^k,\] 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.
Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/a03b4f05d024f2fcf402cf3cc52ed48c86d41ebe
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In Applied Mathematics
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Critical points of random polynomials with independent identically distributed roots (2012, methodology): “A similar method appeared in the study of roots of polynomials whose coefficients (not roots) are independent random variables (see [5]), and in the random matrix theory (see [12]).”