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Central limit theorems for biorthogonal ensembles and asymptotics of recurrence coefficients


Authors: Jonathan Breuer and Maurice Duits
Journal: J. Amer. Math. Soc. 30 (2017), 27-66
MSC (2010): Primary 15B52, 42C05, 60F05
DOI: https://doi.org/10.1090/jams/854
Published electronically: January 28, 2016
MathSciNet review: 3556288
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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
Email: duits@kth.se

DOI: https://doi.org/10.1090/jams/854
Keywords: Central limit theorems, linear statistics, random matrix theory, biorthogonal ensembles, Jacobi matrices, right limits, orthogonal polynomials
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.
Article copyright: © Copyright 2016 American Mathematical Society