Limit theorems 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 xP(x)=JP(x). For polynomial linear statistics of such ensembles, we reformulate the cumulant method introduced in [53] in terms of counting certain lattice paths on the adjacency graph of the recurrence matrix J. In the spirit of [12], we show that the asymptotic fluctuations are described by the right-limits of the matrix J. Moreover, whenever the right-limit is a Laurent matrix, we show 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) and a law of large numbers in the one-cut regime. We also prove a CLT for the square singular values of the product of independent complex rectangular Ginibre matrices, as well as for the Laguerre and Jacobi biorthogonal ensembles introduced in [7], and we explain how to recover the equilibrium measure from the asymptotics of the recurrence coefficients. Finally, we discuss the connection with the Strong Szegő limit theorem where this combinatorial method originates.