Virial expansion for almost diagonal random matrices

Energy level statistics of Hermitian random matrices $\hat H$ with Gaussian independent random entries $H_{i\geq j}$ is studied for a generic ensemble of almost diagonal random matrices with $<|H_{ii}|^{2} > \sim 1$ and $<|H_{i\neq j}|^{2} >= b {\cal F}(|i-j|) \ll 1$. We perform a regular expansion of the spectral form-factor $K(τ) = 1 + b K_{1}(τ) + b^{2} K_{2}(τ) + ...$ in powers of $b \ll 1$ with the coefficients $K_{m}(τ)$ that take into account interaction of (m+1) energy levels. To calculate $K_{m}(τ)$, 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 $K_{1}(τ)$ and $K_{2}(τ)$ in terms of infinite series are found for a generic function ${\cal F}(|i-j|)$ 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.

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