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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1310.5262","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-10-19T18:59:54Z","cross_cats_sorted":[],"title_canon_sha256":"f383c6d611ee38c50e9ea9aaa61950773030602deed893e4df997623305142d7","abstract_canon_sha256":"8fab6990fd27e18fbaed01b4b454e1ff7fa70f3da7f0c71b792ac3f556367fbf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:09:46.258098Z","signature_b64":"cPTPqO/ikDM4DzD3h9fpzRiXr3u1LNeCCHmlXnvdp0DL98aGZpSHaLmI1ACBJdR8gKn8+XUKRPbAldWj4r8xAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"84d72200da1f51b6b70ddbc2e94c5c75658db52de9e5da3e8fddfe3c3a91bfb3","last_reissued_at":"2026-05-18T03:09:46.257238Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:09:46.257238Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\mathbb{Z}^d$. 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