Hole Phenomenon of Gaussian Analytic Functions with Power-exponential Weights
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915887877980160 |
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| author | Du, Yun-Heng |
| author_facet | Du, Yun-Heng |
| contents | We establish the \emph{hole phenomenon} for the Gaussian analytic function \[ F_β(z)=\sum_{n=0}^{\infty}\frac{ξ_{n}}{\sqrt{Γ\bigl(\frac{2}β(n+1)\bigr)}}\,z^{n}, \] associated with the power-exponential weight $e^{-|z|^β}$ on $\mathbb{C}$, where $β>0$. Under the condition that $F_β(z)$ has no zeros in $D(0,r)$, the scaled zero counting measure converges to a limiting measure $μ_{0}^β$ vaguely in distribution. This limit exhibits a \emph{forbidden region} \[ \bigl\{1<|z|<e^{1/β}\bigr\}, \] which zeros asymptotically avoid. This generalizes the remarkable discovery of Ghosh and Nishry for the Gaussian entire function (the case $β=2$), 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_24193 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hole Phenomenon of Gaussian Analytic Functions with Power-exponential Weights Du, Yun-Heng Complex Variables Probability We establish the \emph{hole phenomenon} for the Gaussian analytic function \[ F_β(z)=\sum_{n=0}^{\infty}\frac{ξ_{n}}{\sqrt{Γ\bigl(\frac{2}β(n+1)\bigr)}}\,z^{n}, \] associated with the power-exponential weight $e^{-|z|^β}$ on $\mathbb{C}$, where $β>0$. Under the condition that $F_β(z)$ has no zeros in $D(0,r)$, the scaled zero counting measure converges to a limiting measure $μ_{0}^β$ vaguely in distribution. This limit exhibits a \emph{forbidden region} \[ \bigl\{1<|z|<e^{1/β}\bigr\}, \] which zeros asymptotically avoid. This generalizes the remarkable discovery of Ghosh and Nishry for the Gaussian entire function (the case $β=2$), 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. |
| title | Hole Phenomenon of Gaussian Analytic Functions with Power-exponential Weights |
| topic | Complex Variables Probability |
| url | https://arxiv.org/abs/2602.24193 |