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In both regimes, we establish the asymptotic expansion $$ \\log Z_n = C_1 n^2 + "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2508.00316","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math-ph","submitted_at":"2025-08-01T04:56:36Z","cross_cats_sorted":["math.CV","math.MP","math.PR"],"title_canon_sha256":"18a7e718327f1b6300fa116f88128f5aeab4c77c0c119866c9a0e41f561fdafd","abstract_canon_sha256":"c47186d85acc5bfd23fa8c5430f882c58a5636e92da505c0f3e4e6172e208dfd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:46:53.278719Z","signature_b64":"EcOzYfeMiTdoA3rS4m4jFIl8HI4jZheAoz2pbF0KrdUc/6TRz23JOAigjSz+RhoZqFZGQb5nnRTXLxenZ9KdAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"530ee9b2b36f163d1d854eb7ccaee38788594b3005bb4edd14883ba98313e039","last_reissued_at":"2026-07-05T11:46:53.278167Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:46:53.278167Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Anomalous free energy expansions of planar Coulomb gases: multi-component and conformal singularity","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CV","math.MP","math.PR"],"primary_cat":"math-ph","authors_text":"Sung-Soo Byun","submitted_at":"2025-08-01T04:56:36Z","abstract_excerpt":"We study the partition function $$ Z_n = \\int_{\\mathbb{C}^n } \\prod_{1 \\le j<k \\le n} |z_{j}-z_{k}|^{2} \\prod_{j=1}^{n} |z_j|^{2c}\\, e^{-n V(z_{j})}\\frac{d^{2}z_{j}}{\\pi}, $$ where $c>-1$ and $$ V(z)= |z|^{2d}-t(z^{d}+\\overline{z}^{d}), \\qquad t >0, \\, d \\in \\mathbb{N}. $$ The associated droplet reveals a topological phase transition: for $t > 1/\\sqrt{d}$, it consists of $d$ connected components; whereas for $t < 1/\\sqrt{d}$, it becomes simply connected and contains the origin, where a conformal singularity arises. In both regimes, we establish the asymptotic expansion $$ \\log Z_n = C_1 n^2 + "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.00316","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/2508.00316/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":"2508.00316","created_at":"2026-07-05T11:46:53.278225+00:00"},{"alias_kind":"arxiv_version","alias_value":"2508.00316v1","created_at":"2026-07-05T11:46:53.278225+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2508.00316","created_at":"2026-07-05T11:46:53.278225+00:00"},{"alias_kind":"pith_short_12","alias_value":"KMHOTMVTN4LD","created_at":"2026-07-05T11:46:53.278225+00:00"},{"alias_kind":"pith_short_16","alias_value":"KMHOTMVTN4LD2HMF","created_at":"2026-07-05T11:46:53.278225+00:00"},{"alias_kind":"pith_short_8","alias_value":"KMHOTMVT","created_at":"2026-07-05T11:46:53.278225+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2605.29594","citing_title":"Free energy expansion of determinantal Coulomb gases in the quadratic fields with a point charge","ref_index":29,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/KMHOTMVTN4LD2HMFJ234ZLXDQ6","json":"https://pith.science/pith/KMHOTMVTN4LD2HMFJ234ZLXDQ6.json","graph_json":"https://pith.science/api/pith-number/KMHOTMVTN4LD2HMFJ234ZLXDQ6/graph.json","events_json":"https://pith.science/api/pith-number/KMHOTMVTN4LD2HMFJ234ZLXDQ6/events.json","paper":"https://pith.science/paper/KMHOTMVT"},"agent_actions":{"view_html":"https://pith.science/pith/KMHOTMVTN4LD2HMFJ234ZLXDQ6","download_json":"https://pith.science/pith/KMHOTMVTN4LD2HMFJ234ZLXDQ6.json","view_paper":"https://pith.science/paper/KMHOTMVT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2508.00316&json=true","fetch_graph":"https://pith.science/api/pith-number/KMHOTMVTN4LD2HMFJ234ZLXDQ6/graph.json","fetch_events":"https://pith.science/api/pith-number/KMHOTMVTN4LD2HMFJ234ZLXDQ6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KMHOTMVTN4LD2HMFJ234ZLXDQ6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KMHOTMVTN4LD2HMFJ234ZLXDQ6/action/storage_attestation","attest_author":"https://pith.science/pith/KMHOTMVTN4LD2HMFJ234ZLXDQ6/action/author_attestation","sign_citation":"https://pith.science/pith/KMHOTMVTN4LD2HMFJ234ZLXDQ6/action/citation_signature","submit_replication":"https://pith.science/pith/KMHOTMVTN4LD2HMFJ234ZLXDQ6/action/replication_record"}},"created_at":"2026-07-05T11:46:53.278225+00:00","updated_at":"2026-07-05T11:46:53.278225+00:00"}