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Let \\[\\bar{Z}_{n}^\\beta(\\gamma) = Z_{n}^\\beta(\\gamma)/\\left(2 \\textrm{cap}(\\gamma)\\right)^{\\beta n^2/2+(1-\\beta/2)n}\\] be the normalized partition function. We compute the free energy as the number of particles tends to infinity, including the constant term: \\[ \\lim_{n\\to \\infty}\\log \\frac{\\bar{Z}_{n}^\\beta(\\gamma)}{\\bar{Z}_{n}^\\beta([-1,1])} = \\frac{1}{24}J^A(\\gamma) + \\frac{1}{8} \\left( \\sqrt{ \\frac{\\beta}{2} } - \\sqrt{ \\frac{2}{\\beta}} \\right)^2 J^{F}"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2504.19887","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2025-04-28T15:19:09Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"92d4b8c0ffa69468edf558e16b6438bff51769e86e7a0ad9c431fd5601d250ba","abstract_canon_sha256":"a84ceaf5b869159a8e4606771be8a6619bb97f1cd1da7d8c66e4dbb73c0e86c8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:55:05.579919Z","signature_b64":"H65kDpaK9ymHd31xJ/TwehVPU1b74kNTi/dEsf0DmUjDTzcQk+vmNm4RfWISSlUbRrTVdjstZzpSabrHoWZlAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6e9f58e3035111bb54c34b176607f17d47315d57da92071e154a16bb03d4d912","last_reissued_at":"2026-07-05T10:55:05.579440Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:55:05.579440Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Planar Coulomb gas on a Jordan arc at any temperature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CV","authors_text":"Fredrik Viklund, Klara Courteaut, Kurt Johansson","submitted_at":"2025-04-28T15:19:09Z","abstract_excerpt":"We study a planar Coulomb gas confined to a sufficiently smooth Jordan arc $\\gamma$ in the complex plane, at inverse temperature $\\beta > 0$. Let \\[\\bar{Z}_{n}^\\beta(\\gamma) = Z_{n}^\\beta(\\gamma)/\\left(2 \\textrm{cap}(\\gamma)\\right)^{\\beta n^2/2+(1-\\beta/2)n}\\] be the normalized partition function. We compute the free energy as the number of particles tends to infinity, including the constant term: \\[ \\lim_{n\\to \\infty}\\log \\frac{\\bar{Z}_{n}^\\beta(\\gamma)}{\\bar{Z}_{n}^\\beta([-1,1])} = \\frac{1}{24}J^A(\\gamma) + \\frac{1}{8} \\left( \\sqrt{ \\frac{\\beta}{2} } - \\sqrt{ \\frac{2}{\\beta}} \\right)^2 J^{F}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.19887","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/2504.19887/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2504.19887","created_at":"2026-07-05T10:55:05.579491+00:00"},{"alias_kind":"arxiv_version","alias_value":"2504.19887v1","created_at":"2026-07-05T10:55:05.579491+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2504.19887","created_at":"2026-07-05T10:55:05.579491+00:00"},{"alias_kind":"pith_short_12","alias_value":"N2PVRYYDKEI3","created_at":"2026-07-05T10:55:05.579491+00:00"},{"alias_kind":"pith_short_16","alias_value":"N2PVRYYDKEI3WVGD","created_at":"2026-07-05T10:55:05.579491+00:00"},{"alias_kind":"pith_short_8","alias_value":"N2PVRYYD","created_at":"2026-07-05T10:55:05.579491+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2506.11707","citing_title":"Fluctuations of two-dimensional determinantal processes associated with Berezin--Toeplitz operators","ref_index":33,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/N2PVRYYDKEI3WVGDJMLWMB7RPV","json":"https://pith.science/pith/N2PVRYYDKEI3WVGDJMLWMB7RPV.json","graph_json":"https://pith.science/api/pith-number/N2PVRYYDKEI3WVGDJMLWMB7RPV/graph.json","events_json":"https://pith.science/api/pith-number/N2PVRYYDKEI3WVGDJMLWMB7RPV/events.json","paper":"https://pith.science/paper/N2PVRYYD"},"agent_actions":{"view_html":"https://pith.science/pith/N2PVRYYDKEI3WVGDJMLWMB7RPV","download_json":"https://pith.science/pith/N2PVRYYDKEI3WVGDJMLWMB7RPV.json","view_paper":"https://pith.science/paper/N2PVRYYD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2504.19887&json=true","fetch_graph":"https://pith.science/api/pith-number/N2PVRYYDKEI3WVGDJMLWMB7RPV/graph.json","fetch_events":"https://pith.science/api/pith-number/N2PVRYYDKEI3WVGDJMLWMB7RPV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/N2PVRYYDKEI3WVGDJMLWMB7RPV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/N2PVRYYDKEI3WVGDJMLWMB7RPV/action/storage_attestation","attest_author":"https://pith.science/pith/N2PVRYYDKEI3WVGDJMLWMB7RPV/action/author_attestation","sign_citation":"https://pith.science/pith/N2PVRYYDKEI3WVGDJMLWMB7RPV/action/citation_signature","submit_replication":"https://pith.science/pith/N2PVRYYDKEI3WVGDJMLWMB7RPV/action/replication_record"}},"created_at":"2026-07-05T10:55:05.579491+00:00","updated_at":"2026-07-05T10:55:05.579491+00:00"}