{"paper":{"title":"The critical threshold for Bargmann-Fock percolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alejandro Rivera (IF), Hugo Vanneuville (ICJ, PSPM)","submitted_at":"2017-11-14T09:26:30Z","abstract_excerpt":"In this article, we study the excursions sets $\\mathcal{D}\\_p=f^{-1}([-p,+\\infty[)$ where $f$ is a natural real-analytic planar Gaussian field called the Bargmann-Fock field. More precisely, $f$ is the centered Gaussian field on $\\mathbb{R}^2$ with covariance $(x,y) \\mapsto \\exp(-\\frac{1}{2}|x-y|^2)$. In [BG16], Beffara and Gayet prove that, if $p \\leq 0$, then a.s. $\\mathcal{D}\\_p$ has no unbounded component. We show that conversely, if $p>0$, then a.s. $\\mathcal{D}\\_p$ has a unique unbounded component. As a result, the critical level of this percolation model is $0$. We also prove exponentia"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.05012","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"}