论文标题
向Tracy-Widom分布的平稳过渡，以互动$ k $ - body fermionic嵌入式高斯合奏的最大特征值
A smooth transition towards a Tracy-Widom distribution for the largest eigenvalue of interacting $k$-body fermionic Embedded Gaussian Ensembles
论文作者
论文摘要
尽管它简单，但中心限制定理捕获了数学物理学中最杰出的现象之一，即普遍性。尽管该经典结果众所周知，当随机变量变得相关时，仍然不清楚这种普遍行为会发生什么。随机矩阵的特征值集提供了一个富有成果的数学实验室，以研究新的通用性能的上升。在这方面，使用标准的随机矩阵集合并专注于极端特征值的分布，已经完成了许多工作。在这种情况下，最大或最小的特征值的分布偏离了Fisher-Tippett-gnedenko定理，产生了著名的Tracy-Widom分布。当特征值之间的相关性发生变化时，人们可能会再次想知道这种新的普遍行为是多么健壮。对这个凄美的问题提供了很少的答案，我们在本工作中的意图是为这个有趣的未开发领域做出贡献。因此，我们从数值上研究了相互作用的$ k $ fermionic正交和统一的嵌入式高斯集团的归一化最大特征值的概率分布。我们发现从略微不对称的高斯式分布（对于小$ k/m $）到tracy-widom分布（$ k/m \ to $ k/m \ to $ k/m \ to $ k $），$ k $是交互的等级，而$ m $是费米斯（Fermions）的数量。对于$ k/m $的小值，光谱边缘的相关性更强，并且独立于所考虑的粒子数量。我们的结果表明，朝向光谱边缘的微妙相关性区分了相互作用的多体系统在稀数极限下的统计特性与对标准随机矩阵集合的期望的频谱。
In spite of its simplicity, the central limit theorem captures one of the most outstanding phenomena in mathematical physics, that of universality. While this classical result is well understood it is still not very clear what happens to this universal behaviour when the random variables become correlated. A fruitful mathematically laboratory to investigate the rising of new universal properties is offered by the set of eigenvalues of random matrices. In this regard a lot of work has been done using the standard random matrix ensembles and focusing on the distribution of extreme eigenvalues. In this case, the distribution of the largest -- or smallest -- eigenvalue departs from the Fisher-Tippett-Gnedenko theorem yielding the celebrated Tracy-Widom distribution. One may wonder, yet again, how robust is this new universal behaviour captured by the Tracy-Widom distribution when the correlation among eigenvalues changes. Few answers have been provided to this poignant question and our intention in the present work is to contribute to this interesting unexplored territory. Thus, we study numerically the probability distribution for the normalized largest eigenvalue of the interacting $k$-body fermionic orthogonal and unitary Embedded Gaussian Ensembles in the diluted limit. We find a smooth transition from a slightly asymmetric Gaussian-like distribution, for small $k/m$, to the Tracy-Widom distribution as $k/m\to 1$, where $k$ is the rank of the interaction and $m$ is the number of fermions. Correlations at the edge of the spectrum are stronger for small values of $k/m$, and are independent of the number of particles considered. Our results indicate that subtle correlations towards the edge of the spectrum distinguish the statistical properties of the spectrum of interacting many-body systems in the dilute limit, from those expected for the standard random matrix ensembles.
