Random matrices with external source and multiple orthogonal polynomials
Bleher, P. M. ; Kuijlaars, A. B. J.
arXiv, 0307055 / Harvested from arXiv
Text on arXiv
We show that the average characteristic polynomial P_n(z) = E [\det(zI-M)] of the random Hermitian matrix ensemble Z_n^{-1} \exp(-Tr(V(M)-AM))dM is characterized by multiple orthogonality conditions that depend on the eigenvalues of the external source A. For each eigenvalue a_j of A, there is a weight and P_n has n_j orthogonality conditions with respect to this weight, if n_j is the multiplicity of a_j. The eigenvalue correlation functions have determinantal form, as shown by Zinn-Justin. Here we give a different expression for the kernel. We derive a Christoffel-Darboux formula in case A has two distinct eigenvalues, which leads to a compact formula in terms of a Riemann-Hilbert problem that is satisfied by multiple orthogonal polynomials.
Publié le : 2003-07-28
Classification:  Mathematical Physics,  High Energy Physics - Theory,  Mathematics - Classical Analysis and ODEs 
@article{0307055,
     author = {Bleher, P. M. and Kuijlaars, A. B. J.},
     title = {Random matrices with external source and multiple orthogonal polynomials},
     journal = {arXiv},
     volume = {2003},
     number = {0},
     year = {2003},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0307055}
}

Bleher, P. M.; Kuijlaars, A. B. J. Random matrices with external source and multiple orthogonal polynomials. arXiv, Tome 2003 (2003) no. 0, . http://gdmltest.u-ga.fr/item/0307055/