{"paper":{"title":"Shifted critical threshold in the loop $O(n)$ model at arbitrary small $n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Lorenzo Taggi","submitted_at":"2018-06-25T10:14:15Z","abstract_excerpt":"In the loop $O(n)$ model a collection of mutually-disjoint self-avoiding loops is drawn at random on a finite domain of a lattice with probability proportional to $${\\lambda^{\\# \\mbox{edges}} n^{\\# \\mbox{loops}},}$$ where $\\lambda, n \\in [0, \\infty)$. Let $\\mu$ be the connective constant of the lattice and, for any $n \\in [0, \\infty)$, let $\\lambda_c(n)$ be the largest value of $\\lambda$ such that the loop length admits uniformly bounded exponential moments. It is not difficult to prove that $\\lambda_c(n) =1/\\mu$ when $n=0$ (in this case the model corresponds to the self-avoiding walk) and tha"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.09360","kind":"arxiv","version":4},"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"}