{"paper":{"title":"Level-Spacing Distributions and the Airy Kernel","license":"","headline":"","cross_cats":["cond-mat","math-ph","math.MP","nlin.SI","solv-int"],"primary_cat":"hep-th","authors_text":"Craig A. Tracy, Harold Widom","submitted_at":"1992-12-01T01:21:00Z","abstract_excerpt":"Scaling level-spacing distribution functions in the ``bulk of the spectrum'' in random matrix models of $N\\times N$ hermitian matrices and then going to the limit $N\\to\\infty$, leads to the Fredholm determinant of the sine kernel $\\sin\\pi(x-y)/\\pi (x-y)$. Similarly a scaling limit at the ``edge of the spectrum'' leads to the Airy kernel $[{\\rm Ai}(x) {\\rm Ai}'(y) -{\\rm Ai}'(x) {\\rm Ai}(y)]/(x-y)$. In this paper we derive analogues for this Airy kernel of the following properties of the sine kernel: the completely integrable system of P.D.E.'s found by Jimbo, Miwa, M{\\^o}ri and Sato; the expres"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"hep-th/9211141","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/hep-th/9211141/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"}