{"paper":{"title":"Exponential decay in the loop $O(n)$ model: $n> 1$, $x<\\tfrac{1}{\\sqrt{3}}+\\varepsilon(n)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Alexander Glazman, Ioan Manolescu","submitted_at":"2018-10-26T12:59:37Z","abstract_excerpt":"We show that the loop $O(n)$ model on the hexagonal lattice exhibits exponential decay of loop sizes whenever $n> 1$ and $x<\\tfrac{1}{\\sqrt{3}}+\\varepsilon(n)$, for some suitable choice of $\\varepsilon(n)>0$.\n  It is expected that, for $n \\leq 2$, the model exhibits a phase transition in terms of~$x$, that separates regimes of polynomial and exponential decay of loop sizes. In this paradigm, our result implies that the phase transition for $n \\in (1,2]$ occurs at some critical parameter $x_c(n)$ strictly greater than that $x_c(1) = 1/\\sqrt3$. The value of the latter is known since the loop $O("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.11302","kind":"arxiv","version":2},"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"}