标题： Levy矩阵的层级统计和局域化转变 显示英文标题

标题： Level statistics and localization transitions of Lévy matrices

摘要： 这项工作对Lévy或重尾随机矩阵(LM)进行了深入研究。通过分析resolvent对角元素的概率分布的自洽方程，我们建立了确定局域化转变的方程，并得到了LM的相图。利用基于超对称场论和Dyson布朗运动的论证，我们证明了在非局域化相中，特征值统计与高斯正交系综相同，在局域化相中则为泊松分布。我们的数值计算证实了这些结果，在无限大的LM极限下有效，但也揭示了控制有限尺寸效应的特征尺度在接近转变时发散得比幂律快得多，并且在远离转变时已经非常大。这导致了一个非常宽的过渡区域，在该区域内系统看起来像是处于混合相。我们的结果结合之前获得的结果，现在提供了Lévy矩阵的完整理论。 显示英文摘要

摘要： This work provide a thorough study of L\'evy or heavy-tailed random matrices (LM). By analysing the self-consistent equation on the probability distribution of the diagonal elements of the resolvent we establish the equation determining the localisation transition and obtain the phase diagram of LMs. Using arguments based on super-symmetric field theory and Dyson Brownian motion we show that the eigenvalue statistics is the same one of the Gaussian Orthogonal Ensemble in the whole delocalised phase and is Poisson in the localised phase. Our numerics confirms these findings, valid in the limit of infinitely large LMs, but also reveals that the characteristic scale governing finite size effects diverges much faster than a power law approaching the transition and is already very large far from it. This leads to a very wide cross-over region in which the system looks as if it were in a mixed phase. Our results, together with the ones obtained previously, provide now a complete theory of L\'evy matrices.