Random Sums of Weighted Orthogonal Polynomials in ${\mathbb{C}}^d$

Abstract

We consider random polynomials of the form $G_n(z):={\sum }_{|\hat{I}\pm |\le n}\hat{I}{¾}_{\hat{I}}^{(n)}\pm p_{n,\hat{I}\pm }(z)$ where $\{\hat{I}{¾}_{\hat{I}}^{(n)}\pm {\}}_{|\hat{I}\pm |\le n}$ are i.i.d. (complex) random variables and $\{p_{n,\hat{I}\pm }{\}}_{|\hat{I}\pm |\le n}$ 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 $\{p_{n,\hat{I}\pm }\}$ are orthonormal in a space $L^2(e^{-2nQ}\ddot{I}„)$, where $\ddot{I}„$ is a compactly supported Bernstein-Markov measure and $Q$ is a continuous weight function. Under an optimal moment condition on the random variables $\{\hat{I}{¾}_{\hat{I}}^{(n)}\pm \}$, 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.