Transformations of polynomial ensembles

Arno B. J. Kuijlaars

Published 2015-01-22Version 1

A polynomial ensemble is a probability density function for the position of $n$ real particles of the form $\frac{1}{Z_n} \, \prod_{j<k} (x_k-x_j) \, \det \left[ f_k (x_j) \right]_{j,k=1}^n$, for certain functions $f_1, \ldots, f_n$. Such ensembles appear frequently as the joint eigenvalue density of random matrices. We present a number of transformations that preserve the structure of a polynomial ensemble. These transformations include the restriction of a Hermitian matrix by removing one row and one column, a rank-one modification of a Hermitian matrix, and the extension of a Hermitian matrix by adding an extra row and column with complex Gaussians. A special case of the latter result gives an elementary approach to the joint eigenvalue density of a GUE matrix.