On the Variance of the Index for the Gaussian Unitary Ensemble

2011/10/01 by N. S. Witte, Witte, N. S., P. J. Forrester +1
Mathematics · Physics and Astronomy · #33E17 #60B20 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.CA #math.MP #msc:33E17 #msc:60B20

paper · pdf · doi:10.48550/arxiv.1110.0119

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arxiv created 2011/10/05 · arxiv updated 2011/10/06

Abstract

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 τ-function of the fourth Painlevé 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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