{"paper":{"title":"Asymptotics of the Hankel determinant and orthogonal polynomials arising from the information theory of MIMO systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Dyson's Coulomb fluid approach yields large-n asymptotic expansions for recurrence coefficients, Hankel determinants, and related quantities for orthogonal polynomials with a deformed Laguerre weight from MIMO information theory.","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Chao Min, Xiaoqing Wu","submitted_at":"2025-10-08T07:52:33Z","abstract_excerpt":"We consider the Hankel determinant and orthogonal polynomials with respect to the deformed Laguerre weight $w(x; t) = {x^\\alpha }{\\mathrm e^{ - x}}{(x + t)^\\lambda },\\; x\\in \\mathbb{R}^{+} $ with parameters $\\alpha > -1,\\; t > 0$ and $\\lambda \\in \\mathbb{R}$. This problem originates from the information theory of single-user multiple-input multiple-output (MIMO) systems studied by Chen and McKay [{\\em IEEE Trans. Inf. Theory} {\\bf 58} ({2012}) {4594--4634}]. By using the ladder operators for orthogonal polynomials with general Laguerre-type weights, we obtain a system of difference equations a"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"By using Dyson's Coulomb fluid approach, we obtain the large n asymptotic expansions of the recurrence coefficients α_n(t) and β_n(t), the sub-leading coefficient p(n, t) of the monic orthogonal polynomials, the Hankel determinant D_n(t) and the normalized constant h_n(t) for fixed t∈R+.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That Dyson's Coulomb fluid approach applies directly to the deformed weight w(x;t) and yields the stated leading asymptotics without further corrections for the (x+t)^λ factor or the specific parameter regime (section on large-n analysis).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Derives large-n asymptotics for recurrence coefficients α_n(t), β_n(t), Hankel determinant D_n(t), and related quantities for orthogonal polynomials with weight w(x;t)=x^α e^{-x}(x+t)^λ using ladder operators and Dyson's Coulomb fluid approach, plus long-time asymptotics as t→∞.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Dyson's Coulomb fluid approach yields large-n asymptotic expansions for recurrence coefficients, Hankel determinants, and related quantities for orthogonal polynomials with a deformed Laguerre weight from MIMO information theory.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"20bdb1f31afe95378a86d4003326fe9e55cb2a42a3aa3157e7ccc531963cdcea"},"source":{"id":"2510.06739","kind":"arxiv","version":3},"verdict":{"id":"df3f9e48-ddb3-442e-b067-bdd7e1362d24","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-18T09:31:05.659732Z","strongest_claim":"By using Dyson's Coulomb fluid approach, we obtain the large n asymptotic expansions of the recurrence coefficients α_n(t) and β_n(t), the sub-leading coefficient p(n, t) of the monic orthogonal polynomials, the Hankel determinant D_n(t) and the normalized constant h_n(t) for fixed t∈R+.","one_line_summary":"Derives large-n asymptotics for recurrence coefficients α_n(t), β_n(t), Hankel determinant D_n(t), and related quantities for orthogonal polynomials with weight w(x;t)=x^α e^{-x}(x+t)^λ using ladder operators and Dyson's Coulomb fluid approach, plus long-time asymptotics as t→∞.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That Dyson's Coulomb fluid approach applies directly to the deformed weight w(x;t) and yields the stated leading asymptotics without further corrections for the (x+t)^λ factor or the specific parameter regime (section on large-n analysis).","pith_extraction_headline":"Dyson's Coulomb fluid approach yields large-n asymptotic expansions for recurrence coefficients, Hankel determinants, and related quantities for orthogonal polynomials with a deformed Laguerre weight from MIMO information theory."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2510.06739/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":2,"snapshot_sha256":"07de34b5de3f85b31492352d1660d3492ed3b856ed7fd1ece8c895f19c20c4a3"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}