Hole Phenomenon of Gaussian Analytic Functions with Power-exponential Weights

We establish the \emph{hole phenomenon} for the Gaussian analytic function $$F_{\beta }(z)=\sum _{n=0}^{\mathrm{\infty }}\frac{{\xi }_n}{\sqrt{\mathrm{\Gamma }(\frac{2}{\beta }(n+1))}}{z}^n,$$ associated with the power-exponential weight $e^{-\mathrm{\mid }z{\mathrm{\mid }}^{\beta }}$ 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} $$\{1<\mathrm{\mid }z\mathrm{\mid }<e^{1\mathrm{/}\beta }\},$$ 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.