{"paper":{"title":"Subcritical percolation with a line of defects","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.stat-mech","math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"D. Ioffe, S. Friedli, Y. Velenik","submitted_at":"2011-03-02T12:40:39Z","abstract_excerpt":"We consider the Bernoulli bond percolation process $\\mathbb{P}_{p,p'}$ on the nearest-neighbor edges of $\\mathbb{Z}^d$, which are open independently with probability $p<p_c$, except for those lying on the first coordinate axis, for which this probability is $p'$. Define \\[\\xi_{p,p'}:=-\\lim_{n\\to\\infty}n^{-1}\\log \\mathbb{P}_{p,p'}(0\\leftrightarrow n\\mathbf {e}_1)\\] and $\\xi_p:=\\xi_{p,p}$. We show that there exists $p_c'=p_c'(p,d)$ such that $\\xi_{p,p'}=\\xi_p$ if $p'<p_c'$ and $\\xi_{p,p'}<\\xi_p$ if $p'>p_c'$. Moreover, $p_c'(p,2)=p_c'(p,3)=p$, and $p_c'(p,d)>p$ for $d\\geq 4$. We also analyze the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.0411","kind":"arxiv","version":3},"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"}