A necessary and sufficient condition for convergence of the zeros of random polynomials

Consider random polynomials of the form $G_n = \sum_{i=0}^n \xi_i p_i$, where the $\xi_i$ are i.i.d.\ non-degenerate complex random variables, and $\{p_i\}$ is a sequence of orthonormal polynomials with respect to a regular measure $\tau$ supported on a compact set $K$. We show that the zero measure of $G_n$ converges weakly almost surely to the equilibrium measure of $K$ if and only if $\mathbb{E} \log(1 + |\xi_0|) < \infty$. This generalizes the corresponding result of Ibragimov and Zaporozhets in the case when $p_i(z) = z^i$. We also show that the zero measure of $G_n$ converges weakly in probability to the equilibrium measure of $K$ if and only if $\mathbb{P} (|\xi_0| > e^n) = o(n^{-1})$. Our proofs rely on results from small ball probability and exploit the structure of general orthogonal polynomials. Our methods also work for sequences of asymptotically minimal polynomials in $L^p(\tau)$, where $p \in (0, \infty]$. In particular, sequences of $L^p$-minimal polynomials and (normalized) Faber and Fekete polynomials fall into this class.