Sylvester Index of Random Hermitian Matrices.

The Sylvester index of a random Hermitian matrix in the Gaussian ensemble has been considered by Dean and Majumdar. We consider this Sylvester index for a matrix ensemble of random Hermitian matrices defined by a probability density of the form exp (- tr Q (x))) , where Q is a convex polynomial. The main result is the determination of the statistical distribution of the eigenvalues under the condition of a prescribed Sylvester index. We revisit some known results, giving complete proofs, for which we use logarithmic potential theory and complex analysis. [ABSTRACT FROM AUTHOR]