Virial expansion for almost diagonal random matrices
Abstract
Energy level statistics of Hermitian random matrices hat H with Gaussian independent random entries Higeqj is studied for a generic ensemble of almost diagonal random matrices with langle|Hii|2rangle ~ 1 and langle|Hi\not=j|2rangle = bScript F(|i - j|) ll 1. We perform a regular expansion of the spectral form-factor K(tau) = 1 + bK1(tau) + b2K2(tau) + ... in powers of b ll 1 with the coefficients Km(tau) that take into account interaction of (m + 1) energy levels. To calculate Km(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 K1(tau) and K2(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 Rosenzweig-Porter and power-law banded matrix ensembles are considered as examples.
- Publication:
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Journal of Physics A Mathematical General
- Pub Date:
- August 2003
- DOI:
- 10.1088/0305-4470/36/30/305
- arXiv:
- arXiv:cond-mat/0301395
- Bibcode:
- 2003JPhA...36.8265Y
- Keywords:
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- Condensed Matter - Disordered Systems and Neural Networks;
- Condensed Matter - Mesoscale and Nanoscale Physics;
- Condensed Matter - Statistical Mechanics;
- Mathematical Physics;
- Nuclear Theory;
- Quantum Physics
- E-Print:
- 35 pages, 10 figures, improved abstract and introduction, one more example added, typos corrected, submitted to Journal of Physics A