The probability distributions of the closest neighbour (CN) and farther neighbour (FN) spacings from a given level have been studied for interacting fermion/boson systems with and without spin degree of freedom constructed using an embedded Gaussian orthogonal ensemble (GOE) of one plus random two-body interactions. Our numerical results demonstrate a very good consistency with the recently derived analytical expressions using a 3 × 3 random matrix model and other related quantities by Srivastava et al, J. Phys. A: Math. Theor. 52, 025101 (2019). This establishes conclusively that local level fluctuations generated by embedded ensembles (EE) follow the results of classical Gaussian ensembles.

Separation between average and fluctuation parts in state density in many-particle quantum systems with k-body interactions, modelled by the k-body embedded Gaussian orthogonal random matrices (EGOE(k)), is demonstrated using the method of normal mode decomposition of the spectra and verified using power spectrum analysis, for both fermions and bosons. The smoothed state density is represented by the q-normal distribution (fqN) (with corrections) which is the weight function for q-Hermite polynomials. As the rank of interaction kincreases, the fluctuations set in with smaller order of corrections in the smooth state density. They are found to be of GOE type, for all k values, for both fermion and boson systems.