论文标题
变形随机矩阵的Bose-Einstein样凝结：复制方法
Bose-Einstein-Like condensation of deformed random matrix: A replica approach
论文作者
论文摘要
在这项工作中，我们研究了一个对称变形的随机矩阵，该基质是通过扰动Wigner矩阵的对角线元素而获得的。最小eigenvalue $λ_{\ rm min} $ eigenVector $ \ mathbf {x} _ {\ rm min} $ $λ_ {\ rm min} $的变形随机矩阵的$倾向于在一个站点上凝结。在某些类型的扰动和大组分的极限中，这种凝结变成了尖锐的相变，其机制可以用玻色 - 因斯坦凝结在数学水平上识别。我们通过复制方法研究了这种Bose-Einstein，例如凝结现象。我们首先得出一个公式来计算最小特征值和$ \ mathbf {x} _ {\ rm min} $的统计属性。然后，我们将公式应用于两种可解决的情况：当扰动的分布具有双峰时，并且当它具有连续分布时。对于双峰，我们发现在过渡点，参与率从有限值不连续变为零。相反，在连续分布的情况下，参与率取决于分布，持续或不连续。
In this work, we investigate a symmetric deformed random matrix, which is obtained by perturbing the diagonal elements of the Wigner matrix. The eigenvector $\mathbf{x}_{\rm min}$ of the minimal eigenvalue $λ_{\rm min}$ of the deformed random matrix tends to condensate at a single site. In certain types of perturbations and in the limit of the large components, this condensation becomes a sharp phase transition, the mechanism of which can be identified with the Bose-Einstein condensation in a mathematical level. We study this Bose-Einstein like condensation phenomenon by means of the replica method. We first derive a formula to calculate the minimal eigenvalue and the statistical properties of $\mathbf{x}_{\rm min}$. Then, we apply the formula for two solvable cases: when the distribution of the perturbation has the double peak, and when it has a continuous distribution. For the double peak, we find that at the transition point, the participation ratio changes discontinuously from a finite value to zero. On the contrary, in the case of a continuous distribution, the participation ratio goes to zero either continuously or discontinuously, depending on the distribution.
