Global Fluctuations for Linear Statistics of β-Jacobi Ensembles

2012/03/27 by Ioana Dumitriu, Dumitriu, Ioana, Elliot Paquette +1
Chemistry · Mathematics · #60B20 #Algebraic structures and combinatorial models #FOS: Mathematics #Molecular spectroscopy and chirality #Probability (math.PR) #Random Matrices and Applications #math.PR #msc:60B20

paper · pdf · doi:10.48550/arxiv.1203.6103

43 pages, updated to address other scaling regimes

openalex publication_date 2012/03/27 · arxiv created 2012/10/02 · arxiv updated 2012/10/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the global fluctuations for linear statistics of the form\n\∑_i=1ⁿ f(\λ_i) as n \→ \∞, for C¹ functions f,\nand \λ₁, ..., \λ_n being the eigenvalues of a (general)\n\β-Jacobi ensemble, for which tridiagonal models were given by Killip and\nNenciu as well as Edelman and Sutton. The fluctuation from the mean\n(\∑_i=1ⁿ f(\λ_i) - Exp \∑_i=1ⁿ f(\λ_i)) is given\nasymptotically by a Gaussian process.\n We compute the covariance matrix for the process and show that it is\ndiagonalized by a shifted Chebyshev polynomial basis; in addition, we analyze\nthe deviation from the predicted mean for polynomial test functions, and we\nobtain a law of large numbers.\n

Citations

• Central limit theorem for linear eigenvalue statistics of orthogonally invariant matrix models
• Eigenvalue distributions of large Hermitian matrices; Wigner's semi-circle law and a theorem of Kac, Murdock, and Szegö
• Random-matrix theory of quantum transport
• On the Distribution of the Roots of Certain Symmetric Matrices

Related

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• Beta Jacobi ensembles and associated Jacobi polynomials, II
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• Global spectral fluctuations in the Gaussian Unitary Ensemble