Approximation of plurisubharmonic functions by logarithms of Gaussian analytic functions

Abstract

Let $\Omega$ be a bounded pseudoconvex domain in $\mathbb{C}^N$, and let $u$ be a continuous plurisubharmonic function 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 $n\rightarrow\infty$. Consequently, the normalized zero currents of $f_n$ converge weakly to $dd^c u$. More generally, for every $1\leq k\leq N$, we prove that the normalized currents of simultaneous zeros of $k$ independent copies of $f_n$ converge almost surely to the Bedford--Taylor product $(dd^c u)^k$. We also give a probabilistic proof of the well-known fact that normalized logarithms of the moduli of holomorphic functions are $L^1_{loc}$-dense in the space of plurisubharmonic functions on a pseudoconvex domain.