Virial expansion for almost diagonal random matrices

Energy level statistics of Hermitian random matrices H-circumflex with Gaussian independent random entries Hi≥j is studied for a generic ensemble of almost diagonal random matrices with (vertical bar Hii vertical bar2) ∼ 1 and (vertical bar Hi≠j vertical bar2) bF(vertical bar i - j vertical bar) parallel 1. We perform a regular expansion of the spectral form-factor K(τ) = 1 + bK1(τ) + b2K2(τ) + c in powers of b parallel 1 with the coefficients Km(τ) that take into account interaction of (m + 1) energy levels. To calculate Km(τ), 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(τ) and K2(τ) in terms of infinite series are found for a generic function F(vertical bar i - j vertical bar ) 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