Physics – Mathematical Physics
Scientific paper
2001-06-08
Nucl.Phys. B621 (2002) 643-674
Physics
Mathematical Physics
29 pages, no figures; This is the published version with a few misprints corrected
Scientific paper
10.1016/S0550-3213(01)00508-9
We reconsider the problem of calculating arbitrary negative integer moments of the (regularized) characteristic polynomial for $N\times N$ random matrices taken from the Gaussian Unitary Ensemble (GUE). A very compact and convenient integral representation is found via the use of a matrix integral close to that considered by Ingham and Siegel. We find the asymptotic expression for the discussed moments in the limit of large $N$. The latter is of interest because of a conjectured relation to properties of the Riemann $\zeta-$ function zeroes. Our method reveals a striking similarity between the structure of the negative and positive integer moments which is usually obscured by the use of the Hubbard-Stratonovich transformation. This sheds a new light on "bosonic" versus "fermionic" replica trick and has some implications for the supersymmetry method. We briefly discuss the case of the chiral GUE model from that perspective.
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