Approximation of plurisubharmonic functions by logarithms of Gaussian analytic functions

Abstract

Let $\Omega$ be a bounded pseudoconvex domain in $\mathbb{C}^N$. Given a continuous plurisubharmonic function $u$ on $\Omega$, we construct a sequence of Gaussian analytic functions $f_n$ on $\Omega$ associated with $u$ such that $\frac{1}{n}\log|f_n|$ converges to $u$ in $L^1_{loc}(\Omega)$ almost surely, as $n\rightarrow\infty$. Gaussian analytic function $f_n$ is defined through its covariance, or equivalently, via its reproducing kernel Hilbert space, which corresponds to the weighted Bergman space with weight $e^{-2nu}$ with respect to the Lebesgue measure. As a consequence, we show the normalized zeros of $f_n$ converge to $dd^c u$ in the sense of currents.