Virial expansion for almost diagonal random matrices

Energy level statistics of Hermitian random matrices Ĥ with Gaussian independent random entries H i ≥ j is studied for a generic ensemble of almost diagonal random matrices with ⟨| H ii | 2 ⟩ ∼ 1 and ⟨| H i ≠ j | 2 ⟩ = b (| i − j |) ≪ 1. We perform a regular expansion of the spectral form-factor K (τ) = 1 + bK 1 (τ) + b 2 K 2 (τ) + ⋯ in powers of b ≪ 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 colouring with ( m + 1) colours. Expressions for K 1 (τ) and K 2 (τ) in terms of infinite series are found for a generic function (| 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.