{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/level-spacing-distributions-and-the-airy-1","title":"Level-Spacing Distributions and the Airy Kernel","arxiv_id":"hep-th/9211141","date":"1992-12-01","proceeding":null,"authors":["Craig A. Tracy","Harold Widom"],"abstract":"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 expression, in the case of a single interval, of the Fredholm determinant in terms of a Painlev{\\'e} transcendent; the existence of a commuting differential operator; and the fact that this operator can be used in the derivation of asymptotics, for general $n$, of the probability that an interval contains precisely $n$ eigenvalues.","url_abs":"https://arxiv.org/abs/hep-th/9211141v1","url_pdf":"https://arxiv.org/pdf/hep-th/9211141v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"level-spacing-distributions-and-the-airy-1","repo_url":"https://github.com/danbetea/tracy_widom","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}