Hole Phenomenon of Gaussian Analytic Functions with Power-exponential Weights

Abstract

We establish the \emph{hole phenomenon} for the Gaussian analytic function \[ F_{\beta}(z)=\sum_{n=0}^{\infty}\frac{\xi_{n}}{\sqrt{\Gamma\bigl(\frac{2}{\beta}(n+1)\bigr)}}\,z^{n}, \] associated with the power-exponential weight $e^{-|z|^{\beta}}$ on $\mathbb{C}$, where $\beta>0$. Under the condition that $F_{\beta}(z)$ has no zeros in $D(0,r)$, the scaled zero counting measure converges to a limiting measure $\mu_{0}^{\beta}$ vaguely in distribution. This limit exhibits a \emph{forbidden region} \[ \bigl\{1<|z|<e^{1/\beta}\bigr\}, \] which zeros asymptotically avoid. This generalizes the remarkable discovery of Ghosh and Nishry for the Gaussian entire function (the case $\beta=2$), who first revealed this striking conditional convergence and the emergence of a hole. Our analysis extends their phenomenon to the entire family of power-exponential weights.