On an average over the Gaussian Unitary Ensemble

Details On an average over the Gaussian Unitary Ensemble On an average over the Gaussian Unitary Ensemble

2009-03-31

arxiv.org/abs/0903.5450v1

Int. Math. Res. Notices (2009), Vol. 2009: article ID rnp062, 30 pages

Physics

Mathematical Physics

28 pages, 3 figures

Scientific paper

10.1093/imrn/rnp062

We study the asymptotic limit for large matrix dimension N of the partition function of the unitary ensemble with weight exp(-z^2/2x^2 + t/x - x^2/2). We compute the leading order term of the partition function and of the coefficients of its Taylor expansion. Our results are valid in the range N^(-1/2) < z < N^(1/4). Such partition function contains all the information on a new statistics of the eigenvalues of matrices in the Gaussian Unitary Ensemble (GUE) that was introduced by Berry and Shukla (J. Phys. A: Math. Theor., Vol. 41 (2008), 385202, arXiv:0807.3474). It can also be interpreted as the moment generating function of a singular linear statistics.

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