{"paper":{"title":"The number of open paths in an oriented $\\rho$-percolation model","license":"","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Francis Comets, Marina Vachkovskaia, Serguei Popov","submitted_at":"2007-07-05T15:50:06Z","abstract_excerpt":"We study the asymptotic properties of the number of open paths of length $n$ in an oriented $\\rho$-percolation model. We show that this number is $e^{n\\alpha(\\rho)(1+o(1))}$ as $n \\to \\infty$. The exponent $\\alpha$ is deterministic, it can be expressed in terms of the free energy of a polymer model, and it can be explicitely computed in some range of the parameters. Moreover, in a restricted range of the parameters, we even show that the number of such paths is $n^{-1/2} W e^{n\\alpha(\\rho)}(1+o(1))$ for some nondegenerate random variable $W$. We build on connections with the model of directed "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0707.0818","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"}