Negative moments of characteristic polynomials of random matrices: Ingham–Siegel integral as an alternative to Hubbard–Stratonovich transformation

Abstract

We reconsider the problem of calculating arbitrary negative integer moments of the (regularised) characteristic polynomial for N× 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.