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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Central limit theorems for biorthogonal ensembles and asymptotics of recurrence coefficients
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by Jonathan Breuer and Maurice Duits PDF
J. Amer. Math. Soc. 30 (2017), 27-66 Request permission

Abstract:

We study fluctuations of linear statistics corresponding to smooth functions for certain biorthogonal ensembles. We study those biorthogonal ensembles for which the underlying biorthogonal family satisfies a finite term recurrence and describe the asymptotic fluctuations using right limits of the recurrence matrix. As a consequence, we show that whenever the right limit is a Laurent matrix, a central limit theorem holds. We will also discuss the implications for orthogonal polynomial ensembles. In particular, we obtain a central limit theorem for the orthogonal polynomial ensemble associated with any measure belonging to the Nevai class of an interval. Our results also extend previous results on unitary ensembles in the one-cut case. Finally, we will illustrate our results by deriving central limit theorems for the Hahn ensemble for lozenge tilings of a hexagon and for the Hermitian two matrix model.
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Additional Information
  • Jonathan Breuer
  • Affiliation: Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
  • Email: jbreuer@math.huji.ac.il
  • Maurice Duits
  • Affiliation: Department of Mathematics, Royal Institute of Technology (KTH), Lindstedtsvägen 25, SE-10044 Stockholm, Sweden
  • MR Author ID: 796143
  • Email: duits@kth.se
  • Received by editor(s): October 8, 2013
  • Received by editor(s) in revised form: September 11, 2015, and November 30, 2015
  • Published electronically: January 28, 2016
  • Additional Notes: The first author is supported in part by the US-Israel Binational Science Foundation (BSF) Grant no. 2010348 and by the Israel Science Foundation (ISF) Grant no. 1105/10.
    The second author is supported in part by Grant no. KAW 2010.0063 from the Knut and Alice Wallenberg Foundation and by the Swedish Research Council (VR) Grant no. 2012-3128.
  • © Copyright 2016 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 30 (2017), 27-66
  • MSC (2010): Primary 15B52, 42C05, 60F05
  • DOI: https://doi.org/10.1090/jams/854
  • MathSciNet review: 3556288