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 $L_{loc}^1(\Omega )$ almost surely, as $n\to \infty$. Consequently, the normalized zero currents of $f_n$ converge weakly to $d{d}^{c}u$. More generally, for every $1\le k\le N$, we prove that the normalized currents of simultaneous zeros of $k$ independent copies of $f_n$ converge almost surely to the Bedford--Taylor product $(d{d}^{c}u{)}^k$. We also give a probabilistic proof of the well-known fact that normalized logarithms of the moduli of holomorphic functions are $L_{loc}^1$-dense in the space of plurisubharmonic functions on a pseudoconvex domain.

arXiv comment15 pages. An error in Section 5 of the previous version has been corrected; added new result on convergence of zeros