{"paper":{"title":"A remark on monotonicity in Bernoulli bond Percolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Aldo Procacci, Bernardo N.B. de Lima, R\\'emy Sanchis","submitted_at":"2015-04-24T16:03:15Z","abstract_excerpt":"Consider an anisotropic independent bond percolation model on the $d$-dimensional hypercubic lattice, $d\\geq 2$, with parameter $p$. We show that the two point connectivity function $P_{p}(\\{(0,\\dots,0)\\leftrightarrow (n,0,\\dots,0)\\})$ is a monotone function in $n$ when the parameter $p$ is close enough to 0. Analogously, we show that truncated connectivity function $P_{p}(\\{(0,\\dots,0)\\leftrightarrow (n,0,\\dots,0), (0,\\dots,0)\\nleftrightarrow\\infty\\})$ is also a monotone function in $n$ when $p$ is close to 1."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.06549","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"}