Average characteristic polynomials for multiple orthogonal polynomial ensembles

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.

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Universality of the matrix Airy and Bessel functions at spectral edges of unitary ensembles

Random Matrices Theory and Application ◽

10.1142/s2010326317500101 ◽

2017 ◽

Vol 06 (03) ◽

pp. 1750010 ◽

Cited By ~ 1

Author(s):

M. Bertola ◽

M. Cafasso

Keyword(s):

Matrix Model ◽

Bessel Functions ◽

Airy Function ◽

Characteristic Polynomials ◽

Average Characteristic ◽

Integral Matrix ◽

Hard Edge ◽

Model Matrix ◽

The Matrix ◽

Soft Edge

This paper deals with products and ratios of average characteristic polynomials for unitary ensembles. We prove universality at the soft edge of the limiting eigenvalues’ density, and write the universal limit in function of the Kontsevich matrix model (“matrix Airy function”, as originally named by Kontsevich). For the case of the hard edge, universality is already known. We show that also in this case the universal limit can be expressed as a matrix integral (“matrix Bessel function”) known in the literature as generalized Kontsevich matrix model.

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Hermite-Padé approximations and multiple orthogonal polynomial ensembles

Russian Mathematical Surveys ◽

10.1070/rm2011v066n06abeh004771 ◽

2011 ◽

Vol 66 (6) ◽

pp. 1133-1199 ◽

Cited By ~ 22

Author(s):

Alexander I Aptekarev ◽

Arno Kuijlaars

Keyword(s):

Orthogonal Polynomial ◽

Padé Approximations ◽

Multiple Orthogonal Polynomial ◽

Pade Approximations

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Alpha Adjacency: A generalization of adjacency matrices

Electronic Journal of Linear Algebra ◽

10.13001/1081-3810.3828 ◽

2019 ◽

Vol 35 ◽

pp. 365-375

Author(s):

Matt Hudelson ◽

Judi McDonald ◽

Enzo Wendler

Keyword(s):

Characteristic Polynomial ◽

Adjacency Matrix ◽

Undirected Graph ◽

Arbitrary Field ◽

Characteristic Polynomials ◽

Average Characteristic ◽

Adjacency Matrices ◽

Skew Adjacency Matrix

B. Shader and W. So introduced the idea of the skew adjacency matrix. Their idea was to give an orientation to a simple undirected graph G from which a skew adjacency matrix S(G) is created. The -adjacency matrix extends this idea to an arbitrary field F. To study the underlying undirected graph, the average -characteristic polynomial can be created by averaging the characteristic polynomials over all the possible orientations. In particular, a Harary-Sachs theorem for the average-characteristic polynomial is derived and used to determine a few features of the graph from the average-characteristic polynomial.

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