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

作者：Neuschel Thorsten^*

来源：Random Matrices-Theory and Applications, 2014, 3(1): 1450003.

DOI：10.1142/S2010326314500038

• Asymptotics
• asymptotic distribution of zeros
• Plancherel-Rotach formula
• multivariate saddle point method
• Fuss-Catalan distribution
• Ginibre random matrices
• average characteristic polynomials
• generalized hypergeometric polynomials

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.

• 出版日期2014-1