{"paper":{"title":"Coupling Brownian loop soups and random walk loop soups at all polynomial scales","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Wei Qian","submitted_at":"2026-01-06T13:06:17Z","abstract_excerpt":"Lawler and Trujillo Ferreras constructed a well-known coupling between the Brownian loop soups on $\\mathbb{R}^2$ and the (discrete-time) random walk loop soups on $\\mathbb{Z}^2$ (one rescales the random walk loops by $1/N$, their time parametrizations by $1/(2N^2)$, and lets $N\\to \\infty$), which led to numerous applications. It nevertheless only holds for loops with time length at least $N^{\\theta-2}$ for $\\theta \\in(2/3,2)$. In particular, there is no control on mesoscopic loops with time length less than $N^{-4/3}$ (i.e. roughly diameter less than $N^{-2/3}$). This coupling was subsequently"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2601.02992","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2601.02992/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"}