We consider random polynomials of the form $G_n(z):= \sum_{ |\alpha|\leq n} \xi^{ (n)}_{ \alpha}p_{ n,\alpha}(z)$ where $\{ \xi^{ (n)}_{ \alpha}\}_{ |\alpha|\leq n}$ are i.i.d. (complex) random variables and $\{ p_{ n,\alpha}\}_{ |\alpha|\leq n}$ form a basis for $\mathcal P_n$, the holomorphic polynomials of degree at most $n$ in ${ \mathbb C}^d$. In particular, this includes the setting where $\{ p_{ n,\alpha}\}$ are orthonormal in a space $L^2(e^{ -2n Q} \tau)$, where $\tau$ is a compactly supported Bernstein-Markov measure and $Q$ is a continuous weight function. Under an optimal moment condition on the random variables $\{ \xi^{ (n)}_{ \alpha}\}$, in dimension $d=1$ we prove convergence in probability of the zero measure to the weighted equilibrium measure, and in dimension $d \ge 2$ we prove convergence of zero currents.
Categories: math.PR math.CV