{"paper":{"title":"Hole Phenomenon of Gaussian Analytic Functions with Power-exponential Weights","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CV","authors_text":"Yun-Heng Du","submitted_at":"2026-02-27T17:17:39Z","abstract_excerpt":"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 remar"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2602.24193","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2602.24193/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}