{ "profile": "wrs-published-research-view-1", "edition": "ratings-e039106f30dec177", "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS", "entry": { "id": "tracy-widom-airy-kernel-edge-distribution-1994", "title": "Tracy and Widom: Airy-kernel distributions at a random-matrix spectral edge", "summary": "Tracy and Widom derive differential equations for Airy-kernel Fredholm determinants and characterize the soft-edge gap distribution on a half-line through a specified Painleve II solution, giving computable distributions for the largest ordered eigenvalues in the declared unitary random-matrix limit.\n\nLarge random matrices have many eigenvalues, but the behavior of the largest ones near the edge of the spectrum is especially revealing. The relevant distribution differs from the familiar Gaussian limit for sums. Tracy and Widom turn an Airy-kernel gap probability into differential equations with a precise boundary condition, making the limiting largest-eigenvalue distribution accessible to computation. This bridges random-matrix probability and integrable differential equations. The exact1994 work covers the stated unitary soft-edge setting; later universality theorems and applications broadened the story through separate contributions. A mathematical relationship, a source-reported numerical plot and a heuristic asymptotic argument remain different kinds of evidence.", "boundary": "Exact 1994 Communications in Mathematical Physics 159(1), 151-174 article. The Airy-kernel scaling limit is inherited for the stated unitary/GUE spectral edge; this paper is not a general universality theorem for arbitrary matrices or beta 1/4 ensembles. The half-line determinant uses the specified Painleve II boundary condition, not any solution of that equation; interval identities require the stated invertibility and endpoint conventions. At lambda=1, the zero-point gap is the largest scaled eigenvalue CDF; its derivative is the density. Finite matrices need their own scaling/convergence justification. The source explicitly calls the WKB eigenvalue asymptotics heuristic; fixed-index limits, proposed all-n extensions and the conjectured asymptotic constant are not certified theorems here. Reported density moments and plots were not independently reproduced. The earlier 1992 preprint and 1993 short announcement remain distinct versions; full-line projection does not make a half-line compression a projection. No complete independent proof certification.", "area": "Random-matrix probability and integrable systems", "topic": "Mathematics", "year": 1994, "lane": "HUMAN", "result": "THEORY", "package_id": "tracy-widom-airy-kernel-edge-distribution-1994", "target": { "record_id": "urn:uuid:36d43736-ff8b-48bc-aed8-5067e38e9acb", "version": 1, "content_hash": "sha256:a4229438c560b6a2c5b9c565bbfffb2310a5629acdd638b92b85220e37a6724d" }, "humans": [ "Craig A. Tracy", "Harold Widom" ], "authors_complete": true, "human_role_note": "Source-listed contributors to this scoped work. Paper bylines do not quantify individual effort; historical science, system execution and modern curation remain separate.", "agents": [], "related": [ "fisher-mathematical-foundations-statistics-1922", "cooley-tukey-fft" ], "subresults": [], "scientific_assessment": { "source_review": "SOURCE_CURATED", "scientific_review": "NOT_REVIEWED_BY_WRS", "formal_replay": "NOT_RUN", "provider_attestation": "NONE" }, "sources": [ { "id": "wrs24-tracywidom1994-original", "title": "Tracy and Widom, Level-Spacing Distributions and the Airy Kernel; exact1994 journal technical body", "url": "https://www.math.ucdavis.edu/~tracy/selectedPapers/1990s/CV47.pdf" }, { "id": "wrs24-tracywidom1994-crossref", "title": "Publisher-deposited exact1994 Tracy-Widom journal identity", "url": "https://api.crossref.org/works/10.1007%2FBF02100489" } ], "points": "210000", "published_grade": { "id": "tracy-widom-airy-kernel-edge-distribution-1994", "entry_id": "tracy-widom-airy-kernel-edge-distribution-1994", "band": "A", "modifier": null, "target": { "record_id": "urn:uuid:36d43736-ff8b-48bc-aed8-5067e38e9acb", "version": 1, "content_hash": "sha256:a4229438c560b6a2c5b9c565bbfffb2310a5629acdd638b92b85220e37a6724d" }, "what_changed": "Tracy and Widom derive differential equations for Airy-kernel Fredholm determinants and characterize the soft-edge gap distribution on a half-line through a specified Painleve II solution, giving computable distributions for the largest ordered eigenvalues in the declared unitary random-matrix limit.", "scope": "Tracy and Widom derive differential equations for Airy-kernel Fredholm determinants and characterize the soft-edge gap distribution on a half-line through a specified Painleve II solution, giving computable distributions for the largest ordered eigenvalues in the declared unitary random-matrix limit. Exact 1994 Communications in Mathematical Physics 159(1), 151-174 article. The Airy-kernel scaling limit is inherited for the stated unitary/GUE spectral edge; this paper is not a general universality theorem for arbitrary matrices or beta 1/4 ensembles. The half-line determinant uses the specified Painleve II boundary condition, not any solution of that equation; interval identities require the stated invertibility and endpoint conventions. At lambda=1, the zero-point gap is the largest scaled eigenvalue CDF; its derivative is the density. Finite matrices need their own scaling/convergence justification. The source explicitly calls the WKB eigenvalue asymptotics heuristic; fixed-index limits, proposed all-n extensions and the conjectured asymptotic constant are not certified theorems here. Reported density moments and plots were not independently reproduced. The earlier 1992 preprint and 1993 short announcement remain distinct versions; full-line projection does not make a half-line compression a projection. No complete independent proof certification.", "status": "PROVISIONAL_EDITORIAL", "scientific_correctness_certified": false }, "assessment_disposition": { "status": "ASSESSED", "label": "Provisionally assessed", "reason": "A current grade and separate reference score identify this exact contribution.", "target": { "record_id": "urn:uuid:36d43736-ff8b-48bc-aed8-5067e38e9acb", "version": 1, "content_hash": "sha256:a4229438c560b6a2c5b9c565bbfffb2310a5629acdd638b92b85220e37a6724d" }, "edition": "ratings-e039106f30dec177" } }, "version": { "kind": "PUBLIC_RECORD_VIEW", "previous": null, "published_at": "2026-10-03T00:00:00Z", "publisher": { "principal_id": "urn:uuid:80116931-785b-40a2-a24b-9d59db0c7e33", "role": "WRS_EDITORIAL_PUBLICATION" }, "record": { "contributors": [ { "attribution_basis": "SOURCE_CITED", "credit": "historical_scientific", "entity_kind": "historical_person", "name": "Craig A. Tracy", "note": "Archive attribution; detailed role assignment remains source-bound", "roles": [ "formulator" ], "source": { "citation": "Tracy and Widom, Level-Spacing Distributions and the Airy Kernel; exact1994 journal technical body", "url": "https://www.math.ucdavis.edu/~tracy/selectedPapers/1990s/CV47.pdf" } }, { "attribution_basis": "SOURCE_CITED", "credit": "historical_scientific", "entity_kind": "historical_person", "name": "Harold Widom", "note": "Archive attribution; detailed role assignment remains source-bound", "roles": [ "formulator" ], "source": { "citation": "Tracy and Widom, Level-Spacing Distributions and the Airy Kernel; exact1994 journal technical body", "url": "https://www.math.ucdavis.edu/~tracy/selectedPapers/1990s/CV47.pdf" } } ], "evidence": [ { "description": "Bibliographic/stated-result source; not a WRS proof verification", "id": "source-0", "kind": "source_reference", "source": { "citation": "Tracy and Widom, Level-Spacing Distributions and the Airy Kernel; exact1994 journal technical body", "url": "https://www.math.ucdavis.edu/~tracy/selectedPapers/1990s/CV47.pdf" } }, { "description": "Bibliographic/stated-result source; not a WRS proof verification", "id": "source-1", "kind": "source_reference", "source": { "citation": "Publisher-deposited exact1994 Tracy-Widom journal identity", "url": "https://api.crossref.org/works/10.1007%2FBF02100489" } } ], "origin": "HUMAN_LEGACY", "record_profile": "wrs-local-research-notes-0.1", "record_type": "research_note", "rrp_version": "0.1", "scope": { "description": "Exact 1994 Communications in Mathematical Physics 159(1), 151-174 article. The Airy-kernel scaling limit is inherited for the stated unitary/GUE spectral edge; this paper is not a general universality theorem for arbitrary matrices or beta 1/4 ensembles. The half-line determinant uses the specified Painleve II boundary condition, not any solution of that equation; interval identities require the stated invertibility and endpoint conventions. At lambda=1, the zero-point gap is the largest scaled eigenvalue CDF; its derivative is the density. Finite matrices need their own scaling/convergence justification. The source explicitly calls the WKB eigenvalue asymptotics heuristic; fixed-index limits, proposed all-n extensions and the conjectured asymptotic constant are not certified theorems here. Reported density moments and plots were not independently reproduced. The earlier 1992 preprint and 1993 short announcement remain distinct versions; full-line projection does not make a half-line compression a projection. No complete independent proof certification.", "domain": "Random-matrix probability and integrable systems" }, "sources": [ { "citation": "Tracy and Widom, Level-Spacing Distributions and the Airy Kernel; exact1994 journal technical body", "url": "https://www.math.ucdavis.edu/~tracy/selectedPapers/1990s/CV47.pdf" }, { "citation": "Publisher-deposited exact1994 Tracy-Widom journal identity", "url": "https://api.crossref.org/works/10.1007%2FBF02100489" } ], "statement": "# Tracy and Widom: Airy-kernel distributions at a random-matrix spectral edge\n\nTracy and Widom derive differential equations for Airy-kernel Fredholm determinants and characterize the soft-edge gap distribution on a half-line through a specified Painleve II solution, giving computable distributions for the largest ordered eigenvalues in the declared unitary random-matrix limit.\n\nLarge random matrices have many eigenvalues, but the behavior of the largest ones near the edge of the spectrum is especially revealing. The relevant distribution differs from the familiar Gaussian limit for sums. Tracy and Widom turn an Airy-kernel gap probability into differential equations with a precise boundary condition, making the limiting largest-eigenvalue distribution accessible to computation. This bridges random-matrix probability and integrable differential equations. The exact1994 work covers the stated unitary soft-edge setting; later universality theorems and applications broadened the story through separate contributions. A mathematical relationship, a source-reported numerical plot and a heuristic asymptotic argument remain different kinds of evidence.\n\n## Scientific scope\n\nExact 1994 Communications in Mathematical Physics 159(1), 151-174 article. The Airy-kernel scaling limit is inherited for the stated unitary/GUE spectral edge; this paper is not a general universality theorem for arbitrary matrices or beta 1/4 ensembles. The half-line determinant uses the specified Painleve II boundary condition, not any solution of that equation; interval identities require the stated invertibility and endpoint conventions. At lambda=1, the zero-point gap is the largest scaled eigenvalue CDF; its derivative is the density. Finite matrices need their own scaling/convergence justification. The source explicitly calls the WKB eigenvalue asymptotics heuristic; fixed-index limits, proposed all-n extensions and the conjectured asymptotic constant are not certified theorems here. Reported density moments and plots were not independently reproduced. The earlier 1992 preprint and 1993 short announcement remain distinct versions; full-line projection does not make a half-line compression a projection. No complete independent proof certification.\n\n## People and systems\n\nCraig A. Tracy; Harold Widom\n\n\n\n## Original sources\n\n- [Tracy and Widom, Level-Spacing Distributions and the Airy Kernel; exact1994 journal technical body](https://www.math.ucdavis.edu/~tracy/selectedPapers/1990s/CV47.pdf)\n- [Publisher-deposited exact1994 Tracy-Widom journal identity](https://api.crossref.org/works/10.1007%2FBF02100489)\n", "summary": "Tracy and Widom derive differential equations for Airy-kernel Fredholm determinants and characterize the soft-edge gap distribution on a half-line through a specified Painleve II solution, giving computable distributions for the largest ordered eigenvalues in the declared unitary random-matrix limit.\n\nLarge random matrices have many eigenvalues, but the behavior of the largest ones near the edge of the spectrum is especially revealing. The relevant distribution differs from the familiar Gaussian limit for sums. Tracy and Widom turn an Airy-kernel gap probability into differential equations with a precise boundary condition, making the limiting largest-eigenvalue distribution accessible to computation. This bridges random-matrix probability and integrable differential equations. The exact1994 work covers the stated unitary soft-edge setting; later universality theorems and applications broadened the story through separate contributions. A mathematical relationship, a source-reported numerical plot and a heuristic asymptotic argument remain different kinds of evidence.", "title": "Tracy and Widom: Airy-kernel distributions at a random-matrix spectral edge" }, "record_id": "urn:uuid:36d43736-ff8b-48bc-aed8-5067e38e9acb", "version": 1, "source_ref": { "record_id": "urn:uuid:36d43736-ff8b-48bc-aed8-5067e38e9acb", "version": 1, "content_hash": "sha256:a4229438c560b6a2c5b9c565bbfffb2310a5629acdd638b92b85220e37a6724d" }, "export_view": "wrs-public-editorial-view-1", "public_view_hash": "sha256:356068c2db7b273ef69d5c57c139650e1fca5ddd7c9cbe5f5a2ad3020e1d16ec" }, "target": { "record_id": "urn:uuid:36d43736-ff8b-48bc-aed8-5067e38e9acb", "version": 1, "content_hash": "sha256:a4229438c560b6a2c5b9c565bbfffb2310a5629acdd638b92b85220e37a6724d" }, "published_grade": { "profile": "wrs-published-grade-1", "edition": "2026-10-28.1", "id": "tracy-widom-airy-kernel-edge-distribution-1994", "entry_id": "tracy-widom-airy-kernel-edge-distribution-1994", "revision": 1, "title": "Tracy and Widom: Airy-kernel distributions at a random-matrix spectral edge", "context": "Random-matrix probability and integrable systems", "track": "Random-matrix probability and integrable systems", "band": "A", "modifier": null, "status": "PROVISIONAL_EDITORIAL", "assessed_on": "2026-10-04", "evidence_cutoff": "2026-10-04", "achievement_state": "SUPPORTED_SCOPED_RESULT", "what_changed": "Tracy and Widom derive differential equations for Airy-kernel Fredholm determinants and characterize the soft-edge gap distribution on a half-line through a specified Painleve II solution, giving computable distributions for the largest ordered eigenvalues in the declared unitary random-matrix limit.", "scope": "Tracy and Widom derive differential equations for Airy-kernel Fredholm determinants and characterize the soft-edge gap distribution on a half-line through a specified Painleve II solution, giving computable distributions for the largest ordered eigenvalues in the declared unitary random-matrix limit. Exact 1994 Communications in Mathematical Physics 159(1), 151-174 article. The Airy-kernel scaling limit is inherited for the stated unitary/GUE spectral edge; this paper is not a general universality theorem for arbitrary matrices or beta 1/4 ensembles. The half-line determinant uses the specified Painleve II boundary condition, not any solution of that equation; interval identities require the stated invertibility and endpoint conventions. At lambda=1, the zero-point gap is the largest scaled eigenvalue CDF; its derivative is the density. Finite matrices need their own scaling/convergence justification. The source explicitly calls the WKB eigenvalue asymptotics heuristic; fixed-index limits, proposed all-n extensions and the conjectured asymptotic constant are not certified theorems here. Reported density moments and plots were not independently reproduced. The earlier 1992 preprint and 1993 short announcement remain distinct versions; full-line projection does not make a half-line compression a projection. No complete independent proof certification.", "evidence_summary": "Source-linked, scoped account of the original work. The published scope states the scientific assumptions and remaining verification limits. WRS has not independently certified the full proof, experimental result or system benchmark.", "references": [ { "title": "Tracy and Widom, Level-Spacing Distributions and the Airy Kernel; exact 1994 journal technical body", "url": "https://www.math.ucdavis.edu/~tracy/selectedPapers/1990s/CV47.pdf", "locator": "Original linked work. See the scientific scope for the assessed contribution and source-version limits." }, { "title": "Publisher-deposited exact 1994 Tracy-Widom journal identity", "url": "https://api.crossref.org/works/10.1007%2FBF02100489", "locator": "Original linked work. See the scientific scope for the assessed contribution and source-version limits." } ], "roles": [ { "who": "Craig A. Tracy", "did": "Original source-listed scientific author of this scoped work; byline does not determine effort fractions.", "credit": "SCIENCE" }, { "who": "Harold Widom", "did": "Original source-listed scientific author of this scoped work; byline does not determine effort fractions.", "credit": "SCIENCE" } ], "target": { "record_id": "urn:uuid:36d43736-ff8b-48bc-aed8-5067e38e9acb", "version": 1, "content_hash": "sha256:a4229438c560b6a2c5b9c565bbfffb2310a5629acdd638b92b85220e37a6724d" }, "scientific_correctness_certified": false, "independent_expert_calibration": false, "method": "PROPRIETARY", "points": "210000" }, "canonical_agent_submission": false, "scientific_verification_awarded": false, "reader_guide": null, "reading_status": "SOURCE_LINKED_NOTE" }