Random Sums of Weighted Orthogonal Polynomials in ${\mathbb C}^d$ (2412.11969v1)

Abstract: 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.