CLT for biorthogonal ensembles and related combinatorial identities

We study the fluctuations of certain biorthogonal ensembles for which the underlying family \{P,Q\} satisfies a finite-term recurrence relation of the form $x P(x) = \mathbf{J}P(x)$. For polynomial linear statistics of such ensembles, we reformulate the cumulants' method introduced by Soshnikov in terms of counting lattice paths on the graph of the adjacency matrix \mathbf{J}. In the spirit of Breuer-Duits, we show that the asymptotic fluctuations of polynomial linear statistics are described by the right-limits of the matrix \mathbf{J}. Moreover, whenever the right-limit is a Laurent matrix, we prove that the CLT is equivalent to Soshnikov's main combinatorial lemma. We discuss several applications to unitary invariant Hermitian random matrices. In particular, we provide a general Central Limit Theorem (CLT) in the one-cut regime. We also prove a CLT for square singular values of product of independent complex rectangular Ginibre matrices. Finally, we discuss the connection with the Strong Szeg\H{o} theorem where this combinatorial method originates.