Virial expansion for almost diagonal random matrices
Abstract
Energy level statistics of Hermitian random matrices hat H with Gaussian independent random entries H_{igeqj} is studied for a generic ensemble of almost diagonal random matrices with langleH_{ii}^{2}rangle ~ 1 and langleH_{i\not=j}^{2}rangle = bScript F(i  j) ll 1. We perform a regular expansion of the spectral formfactor K(tau) = 1 + bK_{1}(tau) + b^{2}K_{2}(tau) + ... in powers of b ll 1 with the coefficients K_{m}(tau) that take into account interaction of (m + 1) energy levels. To calculate K_{m}(tau), we develop a diagrammatic technique which is based on the Trotter formula and on the combinatorial problem of graph edges colouring with (m + 1) colours. Expressions for K_{1}(tau) and K_{2}(tau) in terms of infinite series are found for a generic function Script F(i  j) in the Gaussian orthogonal ensemble (GOE), the Gaussian unitary ensemble (GUE) and in the crossover between them (the almost unitary Gaussian ensemble). The RosenzweigPorter and powerlaw banded matrix ensembles are considered as examples.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 August 2003
 DOI:
 10.1088/03054470/36/30/305
 arXiv:
 arXiv:condmat/0301395
 Bibcode:
 2003JPhA...36.8265Y
 Keywords:

 Condensed Matter  Disordered Systems and Neural Networks;
 Condensed Matter  Mesoscale and Nanoscale Physics;
 Condensed Matter  Statistical Mechanics;
 Mathematical Physics;
 Nuclear Theory;
 Quantum Physics
 EPrint:
 35 pages, 10 figures, improved abstract and introduction, one more example added, typos corrected, submitted to Journal of Physics A