Hole Phenomenon of Gaussian Analytic Functions with Power-exponential Weights

We establish the hole phenomenon for the Gaussian analytic function [Formula: see text] associated with the power-exponential weight [Formula: see text] on [Formula: see text], where [Formula: see text]. Under the condition that [Formula: see text] has no zeros in [Formula: see text], the scaled zero counting measure converges to a limiting measure [Formula: see text] vaguely in distribution. This limit exhibits a forbidden region [Formula: see text] which zeros asymptotically avoid. This generalizes the remarkable discovery of Ghosh and Nishry for the Gaussian entire function (the case [Formula: see text]), 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.