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 $k\le m/2$) 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.