Virial expansion for almost diagonal random matrices

Abstract

Energy level statistics of Hermitian random matrices $\hat{H}$ with Gaussian independent random entries $H_{i\ge j}$ is studied for a generic ensemble of almost diagonal random matrices with $<|H_{ii}{|}^2>\sim 1$ and $<|H_{i\ne j}{|}^2>=b\mathcal{F}(|i-j|)\ll 1$. We perform a regular expansion of the spectral form-factor $K(\tau )=1+b{K}_1(\tau )+b^2{K}_2(\tau )+...$ 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 coloring with (m+1) colors. Expressions for $K_1(\tau )$ and $K_2(\tau )$ in terms of infinite series are found for a generic function $\mathcal{F}(|i-j|)$ 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.

35 pages, 10 figures, improved abstract and introduction, one more example added, typos corrected, submitted to Journal of Physics A

References in corpus (1)

• Energy level statistics of a critical random matrix ensemble

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