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 . We compute the leading-order term of the partition function and the coefficients of its Taylor expansion. Our results are valid in the region . 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 .