{"paper":{"title":"Small Brownian Loops Hitting SLE$_2$: An Exact Natural-Content Limit","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Zhengwen Qiao","submitted_at":"2026-07-29T03:37:21Z","abstract_excerpt":"Let $D$ be a bounded analytic Jordan domain and let $\\gamma$ be chordal $\\mathrm{SLE}_2$ in $D$, equipped with its $5/4$-dimensional natural-content measure $\\mu_\\gamma$. We retain the root in the standard integrated Brownian-bridge representation of Brownian loop measure and let $\\mathcal M_\\varepsilon^\\gamma$ be the root-intensity measure of loops with duration in $[\\varepsilon^2,t_0]$ whose traces hit $\\gamma$. For every $f\\in C_c(D)$, we prove that $\\varepsilon^{5/4}\\mathcal M_\\varepsilon^\\gamma(f)$ converges in $L^1$ to $\\frac{4}{5\\pi}\\bar v_{\\mathrm{BB}}\\mu_\\gamma(f)$, where $\\bar v_{\\ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.26439","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/2607.26439/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"}