On an Average over the Gaussian Unitary Ensemble

We study the asymptotic limit for large matrix dimension N of the partition function of the unitary ensemble (β = 2) with weight $w(x):={\mathrm exp}(-\frac {z^2}{2x^2}+\frac {t}{x}-\frac {x^2}{2})$. We compute the leading-order term of the partition function and the coefficients of its Taylor expansion. Our results are valid in the region $c_1\ N^{-\frac {1}{2}}\ <\ z\ <\ c_2\ N^{\frac {1}{4}}$. Such a partition function contains all the information on a new statistics of the eigenvalues of matrices in the Gaussian unitary ensemble that was introduced by Berry and Shukla [2]. It can also be interpreted as the moment-generating function of the singular linear statistics $\Sigma^N_{j=1}\ (1/x_j+1/x^2_j)$.