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This Fredholm determinant describes the critical behavior of the eigenvalue gap probabilities of a random Hermitian matrix chosen from the Unitary Ensemble in the bulk double scaling limit near a quadratic zero of the limiting mean eigenvalue density. Using the Riemann-Hilbert method, we evaluate the large $s$-asympto"},"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":"1209.5415","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-09-24T20:40:25Z","cross_cats_sorted":["math.MP","nlin.SI"],"title_canon_sha256":"27bb8775e6627223961e4837b99fca31f1b4a18e1fdd9030c5f75846f22b3db2","abstract_canon_sha256":"f362260f1accf8c7518b17629f2d15b0ae791a6a57c2a40aa7d6b4f8b62cccb9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:41:40.067836Z","signature_b64":"mhz9kLglfNKVo6rar72M9xTUTEE7sJiLESSzerWNVORfVE2BIgINYwp9uBt1PjOXMC2OarfOF8LhVvl2CG4qBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9836609e7a537aace3b3d9f91d71a8a7e48c0213a72938d8d9ee1289fc76f91c","last_reissued_at":"2026-05-18T03:41:40.067023Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:41:40.067023Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotics of a Fredholm determinant involving the second Painlev\\'e transcendent","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","nlin.SI"],"primary_cat":"math-ph","authors_text":"Alexander Its, Thomas Bothner","submitted_at":"2012-09-24T20:40:25Z","abstract_excerpt":"We study the determinant $\\det(I-K_{\\textnormal{PII}})$ of an integrable Fredholm operator $K_{\\textnormal{PII}}$ acting on the interval $(-s,s)$ whose kernel is constructed out of the $\\Psi$-function associated with the Hastings-McLeod solution of the second Painlev\\'e equation. 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