发表机构
Massachusetts Institute of Technology; New York University（麻省理工学院; 纽约大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究随机首一多项式矩阵行列式零点的极限分布，推广经典圆律、椭圆律和半圆律，提出花瓣律并给出极限密度与支撑的显式公式。
AI 中文摘要
我们研究了 $\det P_N(z)$ 的零点分布，其中 $P_N(z)$ 是一个随机首一多项式矩阵，即 $P_N(z)=z^dI-\sum_{j=0}^{d-1}A_{j,N}z^j$，其中 $A_{j,N}$ 是可能耦合的随机矩阵，其元素方差缩放为 $O(1/N)$。我们给出了在 $N\to\infty$ 时该分布几乎必然弱收敛到一个确定性测度的一般条件，该测度仅依赖于 $A_j$ 元素的方差和协方差。这推广了圆律、椭圆律和半圆律，这些经典律对应 $d=1$ 的特殊情况。与这些经典律不同，这些测度可以将非均匀的二维密度与支撑在曲线上的奇异分量相结合，产生各种花瓣形状的区域，因此我们称之为“花瓣律”。我们给出了极限密度和支撑集的显式公式。在高斯假设下，我们还证明了在极限支撑的小邻域外几乎必然没有特征值。
英文摘要
We study the distribution of the zeros of $\det P_N(z)$ where $P_N(z)$ is a random monic polynomial matrix, i.e., $P_N(z)=z^dI-\sum_{j=0}^{d-1}A_{j,N}z^j$ for possibly coupled random matrices $A_{j,N}$, scaled to have entrywise variance $O(1/N)$. We provide general conditions under which this distribution almost-surely weakly converges as $N\to\infty$ to a deterministic measure, depending on just the variance and covariance of the entries of the $A_j$. This generalizes the circular, elliptic, and semicircle laws, which concern the special case of this question where $d=1$. Unlike those classical laws, these measures can combine nonuniform two-dimensional densities with singular components supported on curves, producing a variety of petal-shaped regions, inspiring our name ``the petaloid law''. We give explicit formulas for the limiting densities and supports. Under a Gaussianity assumption, we also show that there are almost surely no eigenvalues outside small neighborhoods of the limiting support.
