{"paper":{"title":"Loop cluster on the discrete circle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Yinshan Chang","submitted_at":"2013-11-29T14:53:20Z","abstract_excerpt":"The loop clusters of a Poissonian ensemble of Markov loops on a finite or countable graph have been studied in \\cite{Markovian-loop-clusters-on-graphs}. In the present article, we study the loop clusters associated with a rotation invariant nearest neighbor walk on the discrete circle $G^{(n)}$ with $n$ vertices. We prove a convergence result of the loop clusters on $G^{(n)}$, as $n\\rightarrow\\infty$, under suitable condition of the parameters. These parameters are chosen in such a way that the rotation invariant nearest neighbor walk on $G^{(n)}$, as $n\\rightarrow\\infty$, converges to a Brown"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.7583","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"}