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A note on biorthogonal ensembles

Patrick Desrosiers·Peter J. Forrester
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Abstract

We consider ensembles of random matrices, known as biorthogonal ensembles, whose eigenvalue probability density function can be written as a product of two determinants. These systems are closely related to multiple orthogonal functions. It is known that the eigenvalue correlation functions of such ensembles can be written as a determinant of a kernel function. We show that the kernel is itself an average of a single ratio of characteristic polynomials. In the same vein, we prove that the type I multiple polynomials can be expressed as an average of the inverse of a characteristic polynomial. We finally introduce a new biorthogonal matrix ensemble, namely the chiral unitary perturbed by a source term.

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Patrick Desrosiers, Peter J. Forrester. 2006-08-23. A note on biorthogonal ensembles. https://doi.org/10.1016/j.jat.2007.08.006

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