论文标题
非热矩阵数量统计的分析方法
Analytic approach for the number statistics of non-Hermitian random matrices
论文作者
论文摘要
我们介绍了一种强大的分析方法，以研究数量$ \ MATHCAL {n} _ {\ textbf {a}}（γ）$的统计数据，任何轮廓$γ\ in \ mathbb {c} $ in \ mathbb {c} $ for Incriblity Bighty Bighty Bighty tone Bimple-Hermitian north-Hermitian nothermitian nothermitian trician $ {c} $。即使尚不清楚特征值的联合分布的分析表达，我们的通用方法也可以应用于不同的随机矩阵集成。我们说明了具有不对称耦合的加权随机图的邻接矩阵中的方法，为此，标准的随机矩阵工具不可应用。主要结果是一种有效的理论，该理论通过沿$γ$积分的路径积分来确定$ \ Mathcal {n} _ {\ textbf {a}} $的累积生成函数，并从自符号方程的解决方案后进行路径概率分布。我们得出了$ \ Mathcal {n} _ {\ textbf {a}} $的平均值和方差的表达方式，以及速率函数，控制$ {\ Mathcal {n}} _ {\ textbf {\ textbf {a}}}} $ {\ mathcal {n}} _ {将所有理论结果与有限随机矩阵的直接对角线化进行了比较，表现出极好的一致性。
We introduce a powerful analytic method to study the statistics of the number $\mathcal{N}_{\textbf{A}}(γ)$ of eigenvalues inside any contour $γ\in \mathbb{C}$ for infinitely large non-Hermitian random matrices ${\textbf A}$. Our generic approach can be applied to different random matrix ensembles, even when the analytic expression for the joint distribution of eigenvalues is not known. We illustrate the method on the adjacency matrices of weighted random graphs with asymmetric couplings, for which standard random-matrix tools are inapplicable. The main outcome is an effective theory that determines the cumulant generating function of $\mathcal{N}_{\textbf{A}}$ via a path integral along $γ$, with the path probability distribution following from the solution of a self-consistent equation. We derive the expressions for the mean and the variance of $\mathcal{N}_{\textbf{A}}$ as well as for the rate function governing rare fluctuations of ${\mathcal{N}}_{\textbf{A}}{(γ)}$. All theoretical results are compared with direct diagonalization of finite random matrices, exhibiting an excellent agreement.
