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Central Limit Theorems for Biorthogonal Ensembles and Asymptotics of Recurrence Coefficients

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.