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

Consider random polynomials of the form G_n=∑_i=0ⁿξ_ip_i, where the ξ_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 τ 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 Elog⁡(1+|ξ₀|)<∞. This generalizes the corresponding result of Ibragimov and Zaporozhets in the case when p_i(z)=zⁱ. We also show that the zero measure of G_n converges weakly in probability to the equilibrium measure of K if and only if P(|ξ₀|>eⁿ)=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(τ), where p∈(0,∞]. In particular, sequences of L^p-minimal polynomials and (normalized) Faber and Fekete polynomials fall into this class.