标题： 最低特征值在$k$-体玻色随机矩阵系综中的分布 显示英文标题

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

摘要： 我们数值研究了具有$k$-体相互作用的有限多玻色子系统的最低本征值分布，这些相互作用由玻色嵌入高斯正交系综[BEGOE($k$)]和单位系综[BEGUE($k$)]建模。 鉴于最近发表的结果表明，$q$-正则描述了$k$-体嵌入系综的特征值密度的光滑形式，已分析了最低特征值分布的前四个矩作为$q$参数的函数，其中$q \sim 1$对应于$k = 1$，$q = 0$对应于$k = m$；$m$是玻色子的数量。 分析具有挑战性，因为我们处理的是高度相关的变量，但我们为这些分布的质心和方差提供了假设。 这些假设与获得的数值结果非常吻合。 我们的结果显示，分布从接近 1 时类似高斯的 $q$ 平滑过渡到中间值 $q$ 时类似修正的 Gumbel 分布，再到众所周知的 Tracy-Widom 分布（对应于 $q=0$）。 需要强调的是，这是一个新结果，它在数值上证明了具有 $k$-体相互作用的有限多玻色子系统的最低特征值分布从接近 1 时类似高斯的（对应于 $q$），平滑过渡到中间值 $q$ 时类似修正的 Gumbel 分布，再到众所周知的 Tracy-Widom 分布（对应于 $q=0$）。 此外，我们还研究了最低和次低特征值之间的归一化间距分布，发现该分布表现出从 Wigner 猜想（对应于$k=1$）到泊松分布（对应于中间的$k$和$k \le m/2$值）再到 Wigner 猜想（从$k = m/2$到$k = m$）的变化趋势，随着$q$值的减小而变化。 因此，谱边缘处的间距行为与谱体内部的间距不同。 显示英文摘要

摘要： 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 published 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. Analytics are difficult as we are dealing with highly correlated variables, however we provide ansatzs for centroids and variances of these distributions. These match very well with the numerical results obtained. 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 $q=0$. It should be emphasized that this is a new result which numerically demonstrates that the distribution of the lowest eigenvalue of finite many-boson systems with $k$-body interactions 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 $q=0$). In addition, we have also studied the distribution of normalized spacing between the lowest and next lowest eigenvalues and it is seen that this distribution 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 $k = m$) with decreasing $q$ value. Thus, the spacings at the spectrum edge behave differently from the spacings inside the spectrum bulk.