Characteristic polynomials of random Hermitian matrices and Duistermaat-Heckman localisation on non-compact Kähler manifolds. (English) Zbl 0994.15025
Summary: We reconsider the problem of calculating a general spectral correlation function containing an arbitrary number of products and ratios of characteristic polynomials for a \(N\times N\) 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 J. V. Fyodorov [Nucl. Phys. B 621, 643-674 (2002)], is shown to be capable of calculation when reinforced with a generalisation of the Itzykson-Zuber integral to a non-compact integration manifold. We arrive to such a generalisation by discussing the Duistermaat-Heckman localisation principle for integrals over non-compact homogeneous Kähler manifolds. In the limit of large-\(N\) the asymptotic expression for the correlation function reproduces the result outlined earlier by A. V. Andreev and B. D. Simons [Phys. Rev. Lett. 75, 2304 (1995)].
MSC:
| 15B52 | Random matrices (algebraic aspects) |
| 82B41 | Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics |
| 53D20 | Momentum maps; symplectic reduction |
| 58D20 | Measures (Gaussian, cylindrical, etc.) on manifolds of maps |
| 82B44 | Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics |
Keywords:
spectral correlation functions; Gaussian unitary ensemble; Grassmann variable out-integrationReferences:
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