{"paper":{"title":"Critical probability on the product graph of a regular tree and a line","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Kohei Yamamoto","submitted_at":"2018-10-16T17:42:49Z","abstract_excerpt":"We consider Bernoulli bond percolation on the product graph of a regular tree and a line.\n  Schonmann showed that there are a.s. infinitely many infinite clusters at $p=p_u$ by using a certain function $\\alpha(p)$.\n  The function $\\alpha(p)$ is defined by a exponential decay rate of probability that two vertices of the same layer are connected.\n  We show the critical probability $p_c$ can be written by using $\\alpha(p)$.\n  In other words, we construct another definition of the critical probability."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.07162","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"}