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 the 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.