{"paper":{"title":"The total mass of Brownian loop measure of Riemann surfaces for large genus","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.DG","authors_text":"Jiankun Hou, Yunhui Wu","submitted_at":"2026-07-02T13:38:43Z","abstract_excerpt":"Let $\\mathcal{M}_{g,n}(\\mathbf{L})$ be the moduli space of hyperbolic surfaces of genus $g$ with $n \\geq 0$ hyperbolic ends of widths $\\mathbf{L} \\in \\mathbb{R}_{\\geq 0}^n$. We regard the total mass $|\\mu_X^\\kappa|$ of the Brownian loop measure with the killing rate $\\kappa$ as a random variable on $\\mathcal{M}_{g,n}(\\mathbf{L})$. Under the condition $|\\mathbf{L}|^2 =o(g)$ as $g \\to \\infty$, we obtain the following two main results: $(1)$ For any $\\kappa > 0$, the expected value of $|\\mu_X^\\kappa|$ on all non-peripheral homotopy classes over $\\mathcal{M}_{g,n}(\\mathbf{L})$ converges to an expl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.02168","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/2607.02168/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"}