  Let $\xi_0,\xi_1,...$ be independent identically distributed (i.i.d.) random
variables such that $\E \log (1+|\xi_0|)<\infty$. We consider random analytic
functions of the form $$ G_n(z)=\sum_{k=0}^{\infty} \xi_k f_{k,n} z^k, $$ where
$f_{k,n}$ are deterministic complex coefficients. Let $\nu_n$ be the random
measure assigning the same weight $1/n$ to each complex zero of $G_n$. Assuming
essentially that $-\frac 1n \log f_{[tn], n}\to u(t)$ as $n\to\infty$, where
$u(t)$ is some function, we show that the measure $\nu_n$ converges weakly to
some deterministic measure which is characterized in terms of the
Legendre--Fenchel transform of $u$. The limiting measure is universal, that is
it 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.
