Approximation of plurisubharmonic functions by logarithms of Gaussian analytic functions
Kiyoon Eum
TL;DR
The paper addresses approximating a bounded plurisubharmonic function $u$ on a bounded pseudoconvex domain $Ω$ by normalized logarithms of Gaussian analytic functions $f_n$ built from the weighted Bergman spaces $\mathcal{H}(nu)$. By leveraging reproducing kernels and Demailly’s regularization, the authors prove that $\frac{1}{n}\log|f_n|$ converges to $u$ in $L^1_{loc}(Ω)$ almost surely, and that the upper envelope $(\limsup_{n\to\infty} \frac{1}{n}\log|f_n(z)|)^*$ equals $u(z)$ for all $z\in Ω$, with zeros converging to the current $dd^c u$. The framework provides a probabilistic proof of Hörmander’s density result for PSH functions and yields convergence of zero currents, including an explicit description in terms of Bedford–Taylor products. Overall, the work extends prior random-polynomial results to arbitrary continuous psh functions, connecting Gaussian analytic function theory with pluripotential theory and current convergence. The findings offer a stochastic method to approximate plurisubharmonic functions and to study the asymptotic distribution of zeros of holomorphic sections.
Abstract
Let $Ω$ be a bounded pseudoconvex domain in $\mathbb{C}^N$. Given a continuous plurisubharmonic function $u$ on $Ω$, we construct a sequence of Gaussian analytic functions $f_n$ on $Ω$ associated with $u$ such that $\frac{1}{n}\log|f_n|$ converges to $u$ in $L^1_{loc}(Ω)$ 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.