Negative moments of characteristic polynomials of random matrices: Ingham-Siegel integral as an alternative to Hubbard-Stratonovich transformation
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 $ζ-$ 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.