通过识别潜在的Burgers动力学揭示扩散非厄米矩阵中特征向量的重要性
Unveiling the significance of eigenvectors in diffusing non-hermitian matrices by identifying the underlying Burgers dynamics
摘要：
根据我们最近的信，我们详细研究了非厄米特复矩阵的入口扩散。我们得到一个精确的偏微分方程（适用于任何矩阵大小$N$和任意初始条件）的平均扩展特征多项式的发展。这个多项式的对数具有在四元数空间中产生Burgers动力学的势的解释。大$N$中系综的动力学完全由谱密度和某一特征向量关联函数的共同演化决定。这种共同进化是最明显的静电势的四元数参数建立了两个复杂的变量，其中第一个管理标准的光谱特性，而第二解开隐藏的动态特征向量关联函数。我们得到了一般的大$N$公式的谱密度和1点特征向量关联函数适用于任何初始条件。我们通过求解三个例子来验证我们的研究，并通过数值模拟来验证我们的解决方案的解析形式。
英文摘要：
Following our recent letter, we study in detail an entry-wise diffusion of non-hermitian complex matrices. We obtain an exact partial differential equation (valid for any matrix size $N$ and arbitrary initial conditions) for evolution of the averaged extended characteristic polynomial. The logarithm of this polynomial has an interpretation of a potential which generates a Burgers dynamics in quaternionic space. The dynamics of the ensemble in the large $N$ is completely determined by the coevolution of the spectral density and a certain eigenvector correlation function. This coevolution is best visible in an electrostatic potential of a quaternionic argument built of two complex variables, the first of which governs standard spectral properties while the second unravels the hidden dynamics of eigenvector correlation function. We obtain general large $N$ formulas for both spectral density and 1-point eigenvector correlation function valid for any initial conditions. We exemplify our studies by solving three examples, and we verify the analytic form of our solutions with numerical simulations.
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