PLANCHEREL–ROTACH FORMULAE FOR AVERAGE CHARACTERISTIC POLYNOMIALS OF PRODUCTS OF GINIBRE RANDOM MATRICES AND THE FUSS–CATALAN DISTRIBUTION

Abstract

Formulae of Plancherel–Rotach type are established for the average characteristic polynomials of Hermitian products of rectangular Ginibre random matrices on the region of zeros. These polynomials form a general class of multiple orthogonal hypergeometric polynomials generalizing the classical Laguerre polynomials. The proofs are based on a multivariate version of the complex method of saddle points. After suitable rescaling the asymptotic zero distributions for the polynomials are studied and shown to coincide with the Fuss–Catalan distributions. Moreover, introducing appropriate coordinates, elementary and explicit characterizations are derived for the densities as well as for the distribution functions of the Fuss–Catalan distributions of general order.