Distribution of lowest eigenvalue in k-body bosonic random matrix ensembles

We present numerical investigations demonstrating the result that the distribution of the lowest eigenvalue of finite many-boson systems (say we have m number of bosons) with k-body interactions, modeled by Bosonic Embedded Gaussian Orthogonal [BEGOE(k)] and Unitary [BEGUE(k)] random matrix Ensembles of k-body interactions, exhibits a smooth transition from Gaussian like (for k=1) to a modified Gumbel like (for intermediate values of k) to the well-known Tracy–Widom distribution (for k=m) form. We also provide ansatz for centroids and variances of the lowest eigenvalue distributions. In addition, we show that the distribution of normalized spacing between the lowest and next lowest eigenvalues exhibits a transition from Wigner’s surmise (for k=1) to Poisson (for intermediate k values with k≤m/2) to Wigner’s surmise (starting from k=m/2 to k=m) form. We analyze these transitions as a function of q parameter defining q-normal distribution for eigenvalue densities.