Supersymmetry Approach to Almost Diagonal Random Matrices

Details Supersymmetry Approach to Almost Diagonal Random Matrices Supersymmetry Approach to Almost Diagonal Random Matrices

2007-01-18

arxiv.org/abs/cond-mat/0701444v3

J. Phys. A: Math. Theor. 40 (2007) 4691-4716

Physics

Condensed Matter

Disordered Systems and Neural Networks

34 pages, 2 figures, typos corrected

Scientific paper

We develop a supersymmetric field theoretical description of the Gaussian ensemble of the almost diagonal Hermitian Random Matrices. The matrices have independent random entries H_{ij} with parametrically small off-diagonal elements H_{ij}/H_{ii} ~ B << 1. We derive a regular virial expansion of correlation functions in the number of ``interacting'' supermatrices associated with different sites in the real space and demonstrate that the perturbation theory constructed in this way is controlled by a small parameter B. General form of the integral expression for the m-th virial coefficient governed by the ``interaction'' of m supermatrices is presented and calculated explicitly in the cases of 2- and 3-matrix ``interaction''. The suggested technique allows us to calculate both the spectral correlations and the correlations of the eigenfunctions taken at different energies and in different space points.

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