scholarly journals Characteristic Polynomials of Random Matrices

2000 ◽  
Vol 214 (1) ◽  
pp. 111-135 ◽  
Author(s):  
Edouard Brézin ◽  
Shinobu Hikami
Keyword(s):  
Random Matrices ◽  
Characteristic Polynomials

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

2014 ◽  
Vol 03 (01) ◽  
pp. 1450003 ◽  
Author(s):  
THORSTEN NEUSCHEL
Keyword(s):  
Random Matrices ◽  
General Class ◽  
Laguerre Polynomials ◽  
Saddle Points ◽  
Distribution Functions ◽  
Complex Method ◽  
Characteristic Polynomials ◽  
Average Characteristic ◽  
General Order ◽  
Catalan Distribution

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.

scholarly journals Characteristic polynomials of random matrices at edge singularities

2000 ◽  
Vol 62 (3) ◽  
pp. 3558-3567 ◽  
Author(s):  
Edouard Brézin ◽  
Shinobu Hikami
Keyword(s):  
Random Matrices ◽  
Characteristic Polynomials ◽  
Edge Singularities

scholarly journals Random Matrices, Magic Squares and Matching Polynomials

10.37236/1859 ◽  
2004 ◽  
Vol 11 (2) ◽  
Author(s):  
Persi Diaconis ◽  
Alex Gamburd
Keyword(s):  
Random Matrices ◽  
Riemann Zeta Function ◽  
Quantum Chaos ◽  
Higher Moments ◽  
Characteristic Polynomials ◽  
Stochastic Matrices ◽  
Unitary Matrices ◽  
Magic Squares ◽  
Riemann Zeta ◽  
Random Unitary Matrices

Characteristic polynomials of random unitary matrices have been intensively studied in recent years: by number theorists in connection with Riemann zeta-function, and by theoretical physicists in connection with Quantum Chaos. In particular, Haake and collaborators have computed the variance of the coefficients of these polynomials and raised the question of computing the higher moments. The answer turns out to be intimately related to counting integer stochastic matrices (magic squares). Similar results are obtained for the moments of secular coefficients of random matrices from orthogonal and symplectic groups. Combinatorial meaning of the moments of the secular coefficients of GUE matrices is also investigated and the connection with matching polynomials is discussed.

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