{"paper":{"title":"Percolation for the stable marriage of Poisson and Lebesgue","license":"","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Marcelo Ventura Freire, Marina Vachkovskaia, Serguei Popov","submitted_at":"2005-11-07T21:20:05Z","abstract_excerpt":"Let $\\Xi$ be the set of points (we call the elements of $\\Xi$ centers) of Poisson process in $\\R^d$, $d\\geq 2$, with unit intensity. Consider the allocation of $\\R^d$ to $\\Xi$ which is stable in the sense of Gale-Shapley marriage problem and in which each center claims a region of volume $\\alpha\\leq 1$. We prove that there is no percolation in the set of claimed sites if $\\alpha$ is small enough, and that, for high dimensions, there is percolation in the set of claimed sites if $\\alpha<1$ is large enough."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0511186","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"}