CLT for biorthogonal ensembles and related combinatorial identities

Abstract

We study the fluctuations of certain biorthogonal ensembles for which the underlying family \{P,Q\} satisfies a finite-term recurrence relation of the form $xP(x)=\mathbf{J}P(x)$. 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 Szegő theorem where this combinatorial method originates.

45 pages, further explanations added. Version published and available in the author's PhD thesis

References in corpus (5)

• Orthogonal polynomial ensembles in probability theory
• Fluctuations of eigenvalues of matrix models and their applications
• Local universality in biorthogonal Laguerre ensembles
• Gaussian and non-Gaussian fluctuations for mesoscopic linear statistics in determinantal processes
• Muttalib--Borodin ensembles in random matrix theory --- realisations and correlation functions

Cited by in corpus (6)

• Gaussian and non-Gaussian fluctuations for mesoscopic linear statistics in determinantal processes
• Polynomial Ensembles and Recurrence Coefficients
• Incomplete determinantal processes: from random matrix to Poisson statistics
• Relations between generalised Wishart matrices, the Muttalib--Borodin model and matrix spherical functions
• Rate of convergence at the hard edge for various Pólya ensembles of positive definite matrices
• Limiting Empirical Spectral Distribution for Products of Rectangular Matrices