On global universality for zeros of random polynomials

In this work, we study asymptotic zero distribution of random multi-variable polynomials which are random linear combinations Sigma(j) a(j)P(j)(z) with i.i.d coe ffi cients relative to a basis of orthonormal polynomials {P-j}(j) induced by a multi-circular weight function Q defined on C-m satisfying suitable smoothness and growth conditions. In complex dimension m >= 3, we prove that E left perpendicular(log(1 + vertical bar a(j)vertical bar))(m)right perpendicular < infinity is a necessary and sufficient condition for normalized zero currents of random polynomials to be almost surely asymptotic to the (deterministic) extremal current i/pi partial derivative<(partial derivative)over bar>V-Q. In addition, in complex dimension one, we consider random linear combinations of orthonormal polynomials with respect to a regular measure in the sense of Stahl & Totik and we prove analogous results in this setting.