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

TL;DR

This paper develops a novel method for calculating spectral correlation functions of GUE matrices by integrating Grassmann variables early and generalizing the Itzykson-Zuber integral via Duistermaat-Heckman localization on non-compact Kähler manifolds, reproducing known large N results.

Contribution

It introduces a new approach that avoids supersymmetry and Hubbard-Stratonovich transformations, utilizing Duistermaat-Heckman localization for non-compact manifolds to compute spectral correlations.

Findings

Successfully computes spectral correlation functions for GUE matrices.

Reproduces known large N asymptotic results.

Provides a geometric interpretation via localization on Kähler manifolds.

Abstract

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.