Asymptotic distribution of complex zeros of random analytic functions

Let .0,.1,. be independent identically distributed complex-valued random variables such that Elog(1+|.0|)<.. We consider random analytic functions of the formGn(z)=..k=0.kfk,nzk,where fk,n are deterministic complex coefficients. Let .n be the random measure counting the complex zeros of Gn according to their multiplicities. Assuming essentially that .1nlogf[tn],n.u(t) as n.., where u(t) is some function, we show that the measure 1n.n converges in probability to some deterministic measure . which is characterized in terms of the Legendre.Fenchel transform of u. The limiting measure . does not depend on the distribution of the .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.