{"paper":{"title":"Random harmonic maps into spheres","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.OA","math.PR"],"primary_cat":"math.DG","authors_text":"Antoine Song","submitted_at":"2024-02-15T19:30:56Z","abstract_excerpt":"Let $S$ be a punctured Riemann surface with Euler characteristic $\\chi(S)<0$. For any unitary representation $\\rho: \\pi_1(S) \\to U(N)$, we introduce its renormalized energy and its harmonic representatives, which are equivariant harmonic maps from the universal cover of $S$ to the unit sphere in $\\mathbb{C}^N$. Our main result is that if a sequence of unitary representations $\\rho_j$ strongly converges, then their renormalized energies converge to $\\frac{\\pi}{4}|\\chi(S)|$ and the shape of their harmonic representatives converges to a unique rescaled hyperbolic metric. Combining this statement "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.10287","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/2402.10287/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"}