{"paper":{"title":"Tightness of approximations to the chemical distance metric for simple conformal loop ensembles","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math-ph","math.CV","math.MP"],"primary_cat":"math.PR","authors_text":"Jason Miller","submitted_at":"2021-12-15T18:32:01Z","abstract_excerpt":"Suppose that $\\Gamma$ is a conformal loop ensemble (CLE$_\\kappa$) with simple loops ($\\kappa \\in (8/3,4)$) in a simply connected domain $D \\subseteq {\\mathbf C}$ whose boundary is itself a type of CLE$_\\kappa$ loop. Let $\\Upsilon$ be the carpet of $\\Gamma$, i.e., the set of points in $D$ not surrounded by a loop of $\\Gamma$. We prove that certain approximations to the chemical distance metric in $\\Upsilon$ are tight. More precisely, for each path $\\omega \\colon [0,1] \\to \\Upsilon$ and $\\epsilon > 0$ we let ${\\mathfrak N}_\\epsilon(\\omega)$ be the Lebesgue measure of the $\\epsilon$-neighborhood "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2112.08335","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2112.08335/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}