Characteristic polynomials of random Hermitian matrices and Duistermaat-Heckman localisation on non-compact Kaehler manifolds

Details Characteristic polynomials of random Hermitian matrices and Duistermaat-Heckman localisation on non-compact Kaehler manifolds Characteristic polynomials of random Hermitian matrices and Duistermaat-Heckman localisation on non-compact Kaehler manifolds

2002-01-19

arxiv.org/abs/math-ph/0201045v2

Nucl.Phys. B630 (2002) 453-491

Physics

Mathematical Physics

34 page, no figures. In this version we added a few references and modified the introduction accordingly. We also included a n

Scientific paper

We reconsider the problem of calculating a general spectral correlation function containing an arbitrary number of products and ratios of characteristic polynomials for a N x N random matrix taken from the Gaussian Unitary Ensemble (GUE). Deviating from the standard "supersymmetry" approach, we integrate out Grassmann variables at the early stage and circumvent the use of the Hubbard-Stratonovich transformation in the "bosonic" sector. The method, suggested recently by one of us, is shown to be capable of calculation when reinforced with a generalization of the Itzykson-Zuber integral to a non-compact integration manifold. We arrive to such a generalisation by discussing the Duistermaat-Heckman localization principle for integrals over non-compact homogeneous Kaehler manifolds. In the limit of large $N$ the asymptotic expression for the correlation function reproduces the result outlined earlier by Andreev and Simons.

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