We numerically study the distribution of the lowest eigenvalue of finite many-boson systems with $k$-body interactions modeled by Bosonic Embedded Gaussian Orthogonal [BEGOE($k$)] and Unitary [BEGUE($k$)] random matrix Ensembles. Following the recently established result that the $q$-normal describes the smooth form of the eigenvalue density of the $k$-body embedded ensembles, the first four moments of the distribution of lowest eigenvalues have been analyzed as a function of the $q$ parameter, with $q \sim 1$ for $k = 1$ and $q = 0$ for $k = m$; $m$ being the number of bosons. Our results show the distribution exhibits a smooth transition from Gaussian like for $q$ close to 1 to a modified Gumbel like for intermediate values of $q$ to the well-known Tracy-Widom distribution for $q=0$.

翻译：我们数值研究了由玻色子嵌入高斯正交系综[BEGOE($k$)]和幺正系综[BEGUE($k$)]刻画的具有$k$体相互作用的有限多玻色子系统的最低本征值分布。基于近期建立的$q$-正态分布描述$k$体嵌入系综本征值密度平滑形式的结果，我们分析了最低本征值分布的前四阶矩作为参数$q$的函数，其中$k=1$时$q \sim 1$，$k=m$时$q=0$（$m$为玻色子数）。结果表明，该分布呈现从$q$接近1时的高斯型，经中间$q$值时的修正Gumbel型，到$q=0$时著名的Tracy-Widom分布的平滑过渡。