{"corpus_id":6706698,"paper_sha":"e60a89f86e144e867eb9fad83d43c4b52d500c7c","doi":"10.1090/s0025-5718-09-02280-7","arxiv_id":"0804.2543","pmid":null,"pmcid":null,"mag_id":2951868104,"dblp_id":"journals/moc/Bornemann10","acl_id":null,"title":"On the numerical evaluation of Fredholm determinants","year":2008,"publication_date":"2008-04-16","venue":"Mathematics of Computation","journal":{"name":"Math. Comput.","pages":"871-915","volume":"79"},"journal_issn":null,"journal_title":null,"publication_types":["JournalArticle"],"pubmed_pub_types":null,"s2_fields_of_study":["Mathematics","Computer Science"],"reference_count":114,"citation_count":271,"influential_citation_count":36,"is_open_access":true,"arxiv_categories":["math.NA","math.FA"],"arxiv_license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","arxiv_journal_ref":"Math. Comp. 79 (2010) 871-915","mesh_headings":null,"chemicals":null,"comments_corrections":null,"source_flags":1,"s2_open_access_pdf_url":"https://www.ams.org/mcom/2010-79-270/S0025-5718-09-02280-7/S0025-5718-09-02280-7.pdf","s2_open_access_landing_url":"https://www.semanticscholar.org/paper/e60a89f86e144e867eb9fad83d43c4b52d500c7c","s2_open_access_license":"public-domain","s2_open_access_status":"HYBRID","pmc_open_access_pdf_url":null,"pmc_open_access_landing_url":null,"pmc_open_access_license":null,"pmc_open_access_status":null,"unpaywall_open_access_pdf_url":null,"unpaywall_open_access_landing_url":null,"unpaywall_open_access_license":null,"unpaywall_open_access_status":null,"abstract":"Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical treatment of Fredholm determinants to be found in the literature. Instead, the few numerical evaluations that are available rely on eigenfunction expansions of the operator, if expressible in terms of special functions, or on alternative, numerically more straightforwardly accessible analytic expressions, e.g., in terms of Painleve transcendents, that have masterfully been derived in some cases. In this paper we close the gap in the literature by studying projection methods and, above all, a simple, easily implementable, general method for the numerical evaluation of Fredholm determinants that is derived from the classical Nystrom method for the solution of Fredholm equations of the second kind. Using Gauss—Legendre or Clenshaw—Curtis as the underlying quadrature rule, we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel. In particular, we get exponential convergence for analytic kernels, which are typical in random matrix theory. The application of the method to the distribution functions of the Gaussian unitary ensemble (GUE), in the bulk scaling limit and the edge scaling limit, is discussed in detail. After extending the method to systems of integral operators, we evaluate the two-point correlation functions of the more recently studied Airy and Airy 1 processes.","claims":[{"public_id":"cl_a06ee79d08ac00c63f3b2c58c8397a53","status":"active","text":"A simple, easily implementable general method for the numerical evaluation of Fredholm determinants is derived from the classical Nyström method for the solution of Fredholm equations of the second kind.","confidence":0.95,"contributors":[{"id":32,"public_id":"7c402c1b98","public_label":"뀨 (7c402c1b98)","roles":["extraction"],"url":"https://sah.borca.ai/u/7c402c1b98"},{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["review"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/claims/cl_a06ee79d08ac00c63f3b2c58c8397a53"},{"public_id":"cl_c994ae62354722200e08161cba26fe5c","status":"active","text":"After extending the method to systems of integral operators, the two-point correlation functions of the Airy and Airy 1 processes are evaluated.","confidence":0.85,"contributors":[{"id":32,"public_id":"7c402c1b98","public_label":"뀨 (7c402c1b98)","roles":["extraction"],"url":"https://sah.borca.ai/u/7c402c1b98"},{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["review"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/claims/cl_c994ae62354722200e08161cba26fe5c"},{"public_id":"cl_50c8c08df3d87452c4e9175268dc4392","status":"active","text":"Exponential convergence is obtained for analytic kernels, which are typical in random matrix theory.","confidence":0.9,"contributors":[{"id":32,"public_id":"7c402c1b98","public_label":"뀨 (7c402c1b98)","roles":["extraction"],"url":"https://sah.borca.ai/u/7c402c1b98"},{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["review"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/claims/cl_50c8c08df3d87452c4e9175268dc4392"},{"public_id":"cl_bd02c74857a402cf13419a50693ad0ef","status":"active","text":"The method is applied to the distribution functions of the Gaussian unitary ensemble (GUE) in the bulk scaling limit and the edge scaling limit.","confidence":0.85,"contributors":[{"id":32,"public_id":"7c402c1b98","public_label":"뀨 (7c402c1b98)","roles":["extraction"],"url":"https://sah.borca.ai/u/7c402c1b98"},{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["review"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/claims/cl_bd02c74857a402cf13419a50693ad0ef"},{"public_id":"cl_6a60a855079d8f4da9a83fc70df21c61","status":"active","text":"Using Gauss–Legendre or Clenshaw–Curtis as the underlying quadrature rule, the approximation error essentially behaves like the quadrature error for the sections of the kernel.","confidence":0.9,"contributors":[{"id":32,"public_id":"7c402c1b98","public_label":"뀨 (7c402c1b98)","roles":["extraction"],"url":"https://sah.borca.ai/u/7c402c1b98"},{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["review"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/claims/cl_6a60a855079d8f4da9a83fc70df21c61"}],"concepts":[{"public_id":"co_2674aed6f6fa71ac54983faca80db587","status":"active","name":"Fredholm determinants","description":"Determinants of integral operators that express significant quantities in mathematics and physics, notably distribution functions in random matrix theory.","types":["mathematical object"],"aliases":[],"contributors":[{"id":32,"public_id":"7c402c1b98","public_label":"뀨 (7c402c1b98)","roles":["extraction"],"url":"https://sah.borca.ai/u/7c402c1b98"},{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["review"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_2674aed6f6fa71ac54983faca80db587"},{"public_id":"co_28f52adbc6947117becb68ca637aa65b","status":"active","name":"exponential convergence","description":"The convergence rate achieved by the method for analytic kernels, meaning the error decays exponentially with the number of quadrature nodes.","types":["property"],"aliases":[],"contributors":[{"id":32,"public_id":"7c402c1b98","public_label":"뀨 (7c402c1b98)","roles":["extraction"],"url":"https://sah.borca.ai/u/7c402c1b98"},{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["review"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_28f52adbc6947117becb68ca637aa65b"},{"public_id":"co_35461e8ae01df23cc54471ce37c2d425","status":"active","name":"Gauss–Legendre quadrature","description":"A quadrature rule based on Legendre polynomials, used as an underlying quadrature rule for the numerical evaluation method.","types":["method"],"aliases":[],"contributors":[{"id":32,"public_id":"7c402c1b98","public_label":"뀨 (7c402c1b98)","roles":["extraction"],"url":"https://sah.borca.ai/u/7c402c1b98"},{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["review"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_35461e8ae01df23cc54471ce37c2d425"},{"public_id":"co_5cfb17bd2fda90b1d609bfda7f0c2ca5","status":"active","name":"edge scaling limit","description":"The scaling limit applied to the Gaussian unitary ensemble at the edge of the spectrum, where the method is applied.","types":["limit"],"aliases":[],"contributors":[{"id":32,"public_id":"7c402c1b98","public_label":"뀨 (7c402c1b98)","roles":["extraction"],"url":"https://sah.borca.ai/u/7c402c1b98"},{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["review"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_5cfb17bd2fda90b1d609bfda7f0c2ca5"},{"public_id":"co_65b153e00902bba5890720ff69ee4f56","status":"active","name":"random matrix theory","description":"A field of mathematics and physics where distribution functions are naturally expressed as Fredholm determinants, providing typical analytic kernels.","types":["field"],"aliases":[],"contributors":[{"id":32,"public_id":"7c402c1b98","public_label":"뀨 (7c402c1b98)","roles":["extraction"],"url":"https://sah.borca.ai/u/7c402c1b98"},{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["review"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_65b153e00902bba5890720ff69ee4f56"},{"public_id":"co_662674fdb986b7422832035169aedee5","status":"active","name":"Airy process","description":"A stochastic process describing the limiting fluctuations at the edge of the Gaussian unitary ensemble, whose two-point correlation functions are evaluated.","types":["process"],"aliases":[],"contributors":[{"id":32,"public_id":"7c402c1b98","public_label":"뀨 (7c402c1b98)","roles":["extraction"],"url":"https://sah.borca.ai/u/7c402c1b98"},{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["review"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_662674fdb986b7422832035169aedee5"},{"public_id":"co_6b450836572f565e65c4ff883dc226c6","status":"active","name":"two-point correlation functions","description":"Correlation functions evaluated for the Airy and Airy 1 processes after extending the method to systems of integral operators.","types":["function"],"aliases":[],"contributors":[{"id":32,"public_id":"7c402c1b98","public_label":"뀨 (7c402c1b98)","roles":["extraction"],"url":"https://sah.borca.ai/u/7c402c1b98"},{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["review"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_6b450836572f565e65c4ff883dc226c6"},{"public_id":"co_84e5be1af66c8e90736d3f683b6c7919","status":"active","name":"Airy 1 process","description":"A related stochastic process, also studied for its two-point correlation functions using the extended method.","types":["process"],"aliases":[],"contributors":[{"id":32,"public_id":"7c402c1b98","public_label":"뀨 (7c402c1b98)","roles":["extraction"],"url":"https://sah.borca.ai/u/7c402c1b98"},{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["review"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_84e5be1af66c8e90736d3f683b6c7919"},{"public_id":"co_c8549e30d98b25a267d6f9a378b17c73","status":"active","name":"bulk scaling limit","description":"The scaling limit applied to the Gaussian unitary ensemble in the bulk of the spectrum, where the method is applied.","types":["limit"],"aliases":[],"contributors":[{"id":32,"public_id":"7c402c1b98","public_label":"뀨 (7c402c1b98)","roles":["extraction"],"url":"https://sah.borca.ai/u/7c402c1b98"},{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["review"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_c8549e30d98b25a267d6f9a378b17c73"},{"public_id":"co_ca5944ddfd94d99b6911f40b95c0a03e","status":"active","name":"Clenshaw–Curtis quadrature","description":"A quadrature rule based on Chebyshev polynomials, used as an alternative underlying quadrature rule for the numerical evaluation method.","types":["method"],"aliases":[],"contributors":[{"id":32,"public_id":"7c402c1b98","public_label":"뀨 (7c402c1b98)","roles":["extraction"],"url":"https://sah.borca.ai/u/7c402c1b98"},{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["review"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_ca5944ddfd94d99b6911f40b95c0a03e"},{"public_id":"co_cd138e4eae772612eb74b4957cf98ecb","status":"active","name":"Gaussian unitary ensemble","description":"An ensemble of Hermitian random matrices with a Gaussian distribution, whose distribution functions are evaluated using the method in the bulk and edge scaling limits.","types":["mathematical object"],"aliases":["GUE"],"contributors":[{"id":32,"public_id":"7c402c1b98","public_label":"뀨 (7c402c1b98)","roles":["extraction"],"url":"https://sah.borca.ai/u/7c402c1b98"},{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["review"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_cd138e4eae772612eb74b4957cf98ecb"},{"public_id":"co_cf1ee059e2b60483372b6d3c53b4c45f","status":"active","name":"Nyström method","description":"A classical numerical method for solving Fredholm integral equations of the second kind, used here as the basis for evaluating Fredholm determinants.","types":["method"],"aliases":[],"contributors":[{"id":32,"public_id":"7c402c1b98","public_label":"뀨 (7c402c1b98)","roles":["extraction"],"url":"https://sah.borca.ai/u/7c402c1b98"},{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["review"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_cf1ee059e2b60483372b6d3c53b4c45f"},{"public_id":"co_d78ce76ccb962a30e7d90dd81affc86b","status":"active","name":"analytic kernels","description":"Kernels of integral operators that are analytic functions, typical in random matrix theory, for which exponential convergence of the method is proven.","types":["property"],"aliases":[],"contributors":[{"id":32,"public_id":"7c402c1b98","public_label":"뀨 (7c402c1b98)","roles":["extraction"],"url":"https://sah.borca.ai/u/7c402c1b98"},{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["review"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_d78ce76ccb962a30e7d90dd81affc86b"},{"public_id":"co_de5920d6f847027826744e315d1a97c6","status":"active","name":"quadrature error","description":"The error introduced by the quadrature approximation, which the approximation error of the Fredholm determinant essentially follows.","types":["error measure"],"aliases":[],"contributors":[{"id":32,"public_id":"7c402c1b98","public_label":"뀨 (7c402c1b98)","roles":["extraction"],"url":"https://sah.borca.ai/u/7c402c1b98"},{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["review"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_de5920d6f847027826744e315d1a97c6"}],"external_ids":{"DOI":"10.1090/s0025-5718-09-02280-7","ArXiv":"0804.2543","PubMed":null,"PubMedCentral":null,"MAG":2951868104,"DBLP":"journals/moc/Bornemann10","ACL":null},"open_access":{"is_open_access":true,"pdf_url":"https://www.ams.org/mcom/2010-79-270/S0025-5718-09-02280-7/S0025-5718-09-02280-7.pdf","landing_url":"https://www.semanticscholar.org/paper/e60a89f86e144e867eb9fad83d43c4b52d500c7c","source":"semantic_scholar","pdf_url_source":"semantic_scholar_open_access_pdf","license":"public-domain","status":"HYBRID","reason":null},"reference_availability":{"status":"available","references_indexed":true,"full_text_available":true,"full_text_source":"arxiv","count_basis":"semantic_scholar_metadata","extraction_status":"not_applicable","reason":null},"source":{"provider":"episteme2","base_corpus":"semantic_scholar_dump","freshness_mode":"unknown","basis":["semantic_scholar_metadata","postgres_metadata"],"limits":["paper metadata is based on indexed upstream scholarly datasets","claims and concepts are available only for extracted papers","absence of claims or concepts means no extracted graph data is available in this response"],"status":"available","degraded":false,"degraded_reasons":[],"diagnostics":{"status":"available","degraded":false,"degraded_reasons":[],"metadata_status":"available","graph_status":"available","abstract_status":"available"},"source_flags":1},"paper_id":633712,"paper_uid":"d64128fd-6109-4fcb-a773-671649ab0ef1","canonical_identity":{"paper_id":633712,"paper_uid":"d64128fd-6109-4fcb-a773-671649ab0ef1","identity_status":"available","lookup_basis":"semantic_scholar_external_id","compatibility_path":"corpus_id"},"url":"https://sah.borca.ai/papers/6706698"}