{"paper":{"title":"Positivity of the time constant in a continuous model of first passage percolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jean-Baptiste Gou\\'er\\'e, Marie Th\\'eret (LPMA)","submitted_at":"2016-10-19T07:59:32Z","abstract_excerpt":"We consider a non trivial Boolean model $\\Sigma$ on ${\\mathbb R}^d$ for $d\\geq 2$. For every $x,y \\in {\\mathbb R}^d$ we define $T(x,y)$ as the minimum time needed to travel from $x$ to $y$ by a traveler that walks at speed $1$ outside $\\Sigma$ and at infinite speed inside $\\Sigma$. By a standard application of Kingman sub-additive theorem, one easily shows that $T(0,x)$ behaves like $\\mu \\|x\\|$ when $\\|x\\|$ goes to infinity, where $\\mu$ is a constant named the time constant in classical first passage percolation. In this paper we investigate the positivity of $\\mu$. More precisely, under an al"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.05901","kind":"arxiv","version":2},"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"}