{"paper":{"title":"Hole probabilities for finite and infinite Ginibre ensembles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Kartick Adhikari, Nanda Kishore Reddy","submitted_at":"2016-04-28T10:06:30Z","abstract_excerpt":"We study the hole probabilities of the infinite Ginibre ensemble ${\\mathcal X}_{\\infty}$, a determinantal point process on the complex plane with the kernel $\\mathbb K(z,w)= \\frac{1}{\\pi}e^{z\\bar w-\\frac{1}{2}|z|^2-\\frac{1}{2}|w|^2}$ with respect to the Lebesgue measure on the complex plane. Let $U$ be an open subset of open unit disk $\\mathbb D$ and ${\\mathcal X}_{\\infty}(rU)$ denote the number of points of ${\\mathcal X}_{\\infty}$ that fall in $rU$. Then, under some conditions on $U$, we show that $$ \\lim_{r\\to \\infty}\\frac{1}{r^4}\\log\\mathbb P[\\mathcal X_{\\infty}(rU)=0]=R_{\\emptyset}-R_{U}, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.08363","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":""},"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"}