Beta Laguerre ensembles in global regime

Hoang Dung Trinh, Khanh Duy Trinh

Research output: Contribution to journalArticlepeer-review

Abstract

Beta Laguerre ensembles, generalizations of Wishart and Laguerre ensembles, can be realized as eigenvalues of certain random tridiagonal matrices. Analogous to the Wishart (β = 1) case and the Laguerre (β = 2) case, for fixed β, it is known that the empirical distribution of the eigenvalues of the ensembles converges weakly to Marchenko–Pastur distributions, almost surely. The paper restudies the limiting behavior of the empirical distribution but in regimes where the parameter β is allowed to vary as a function of the matrix size N . We show that the above Marchenko–Pastur law holds as long as βN → ∞. When βN → 2c ∈ (0, ∞), the limiting measure is related to associated Laguerre orthogonal polynomials. Gaussian fluctuations around the limit are also studied.

Original languageEnglish
JournalUnknown Journal
Publication statusPublished - 2019 Jul 29

Keywords

  • Associated Laguerre orthogonal polynomials
  • Beta Laguerre ensembles
  • Marchenko–Pastur distributions
  • Poincaré inequality

ASJC Scopus subject areas

  • General

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