On the Variance of the Index for the Gaussian Unitary Ensemble (GUE)
by Dr N. S. Witte
Abstract: Continuing the recent theme of hypergeometric function evaluations of statistical mechanical averages we present an example of an evaluation arising in random matrix theory. The variance of the index is the variance of the distribution in the number of positive eigenvalues of a complex, hermitian matrix with Gaussian iid entries (GUE). The derivation relies on the fact that the generating function for the distribution of the number of positive eigenvalues of the GUE is a \(\tau\)-function of the fourth Painleve equation, and as a consequence of the theory developed for these by the present authors we deduce simple linear, inhomogeneous recurrences for the variance, a simple summation formula, several integral representations and finally an exact hypergeometric function evaluation.
This is joint work with P. Forrester.