{"paper":{"title":"Embedding binary sequences into Bernoulli site percolation on $\\mathbb{Z}^3$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Bernardo N. B. de Lima, Marcelo R. Hil\\'ario, Pierre Nolin, Vladas Sidoravicius","submitted_at":"2013-10-19T18:59:54Z","abstract_excerpt":"We investigate the problem of embedding infinite binary sequences into Bernoulli site percolation on $\\mathbb{Z}^d$ with parameter $p$, known also as percolation of words.\\ In 1995, I.\\ Benjamini and H.\\ Kesten proved that, for $d \\geq 10$ and $p=1/2$, all sequences can be embedded, almost surely. They conjectured that the same should hold for $d \\geq 3$. In this paper we consider $d \\geq 3$ and $p \\in (p_c(d), 1-p_c(d))$, where $p_c(d)<1/2$ is the critical threshold for site percolation on $\\mathbb{Z}^d$. We show that there exists an integer $M = M (p)$, such that, a.s., every binary sequence"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.5262","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"}