{"paper":{"title":"Sharpness of the phase transition and lower bounds for the critical intensity in continuum percolation on $\\mathbb{R}^d$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.stat-mech"],"primary_cat":"math.PR","authors_text":"Sebastian Ziesche","submitted_at":"2016-07-21T07:00:11Z","abstract_excerpt":"We consider the Boolean model $Z$ on $\\mathbb{R}^d$ with random compact grains, i.e. $Z := \\bigcup_{i \\in \\mathbb{N}} (X_i + Z_i)$ where $\\eta_t := \\{X_1, X_2, \\dots\\}$ is a Poisson point process of intensity $t$ and $(Z_1, Z_2, \\dots)$ is an i.i.d. sequence of compact grains (not necessarily balls). We will show, that the volume and diameter of the cluster of a typical grain in $Z$ have an exponential tail if the diameter of the typical grain is a.s. bounded by some constant. To achieve this we adapt the arguments of \\cite{duminil2015newproof} and apply a new construction of the cluster of th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.06211","kind":"arxiv","version":1},"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"}