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 + bK₁(τ) + b²K₂(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 K₁(τ) and K₂(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.