{"paper":{"title":"Matrix Group Integrals, Surfaces, and Mapping Class Groups II: $\\mathrm{O}\\left(n\\right)$ and $\\mathrm{Sp}\\left(n\\right)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.GR","math.MP","math.PR"],"primary_cat":"math.GT","authors_text":"Doron Puder, Michael Magee","submitted_at":"2019-04-30T08:56:47Z","abstract_excerpt":"Let $w$ be a word in the free group on $r$ generators. The expected value of the trace of the word in $r$ independent Haar elements of $\\mathrm{O}(n)$ gives a function ${\\cal T}r_{w}^{\\mathrm{O}}(n)$ of $n$. We show that ${\\cal T}r_{w}^{\\mathrm{O}}(n)$ has a convergent Laurent expansion at $n=\\infty$ involving maps on surfaces and $L^{2}$-Euler characteristics of mapping class groups associated to these maps. This can be compared to known, by now classical, results for the GUE and GOE ensembles, and is similar to previous results concerning $\\mathrm{U}\\left(n\\right)$, yet with some surprising "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.13106","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/1904.13106/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"}