# On the Variance of the Index for the Gaussian Unitary Ensemble

@article{Witte2011OnTV,
title={On the Variance of the Index for the Gaussian Unitary Ensemble},
author={Nicholas S. Witte and Peter J. Forrester},
journal={arXiv: Classical Analysis and ODEs},
year={2011}
}
• Published 1 October 2011
• Mathematics
• arXiv: Classical Analysis and ODEs
We derive simple linear, inhomogeneous recurrences for the variance of the index by utilising the fact that the generating function for the distribution of the number of positive eigenvalues of a Gaussian unitary ensemble is a $\tau$-function of the fourth Painlev\'e equation. From this we deduce a simple summation formula, several integral representations and finally an exact hypergeometric function evaluation for the variance.
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