论文标题
更高的复杂维确定点过程的边缘行为
Edge Behavior of Higher Complex-Dimensional Determinantal Point Processes
论文作者
论文摘要
正如Hedenmalm和Wennman最近证明的那样，这是复杂的随机正常矩阵模型的普遍行为，随着矩阵尺寸的增加，在液滴的边界（也称为边缘）处的互补误差函数行为。在特征值的密度和相关内核中都可以看到这种行为，而Faddeeva等离子体核出现。这些结果在边缘的外部单位正常向量的帮助下整齐地表达。我们证明，这种普遍的行为超越了这类随机正常矩阵，在特定的``椭圆形''类别的确定点过程中也有效，这些过程在$ \ mathbb c^d $上定义，这些过程是确定点过程的较高维点概括，描述了复杂的吉尼布尔eLlept Ellipt Ellipt Ellipt Ellipt Ellipt Ellipt Ellipt Ellipt Ellipt Ellipt eymemble和Complecter Ellipt的特征。这些模型描述了$ \ mathbb c^d $中的粒子系统，并以相互扰动为互助，该系统被外部字段$ \ mathscr v（z）= | z | |^2 - τ\ peripatorAtorname {re}（z_1^2+\ ldots+z_d^2）$，其中$ 0 \ leq fe fe。它们的平均颗粒密度在$ 2D $维椭圆形区域收敛到统一法律。在该区域的高层索上，我们找到了互补的误差函数行为和Faddeeva等离子体内核。据我们所知，这是Faddeeva等离子体内核在更高维模型中出现的第一个实例。结果为$ \ mathbb c^d $上的确定点过程提供了可能的边缘普遍性定理的证据。
As recently proved in generality by Hedenmalm and Wennman, it is a universal behavior of complex random normal matrix models that one finds a complementary error function behavior at the boundary (also called edge) of the droplet as the matrix size increases. Such behavior is seen both in the density of the eigenvalues, and the correlation kernel, where the Faddeeva plasma kernel emerges. These results are neatly expressed with the help of the outward unit normal vector on the edge. We prove that such universal behaviors transcend this class of random normal matrices, being also valid in a specific ``elliptic'' class of determinantal point processes defined on $\mathbb C^d$, which are higher dimensional generalizations of the determinantal point processes describing the eigenvalues of the complex Ginibre ensemble and the complex elliptic Ginibre ensemble. These models describe a system of particles in $\mathbb C^d$ with mutual repulsion, that are confined to the origin by an external field $\mathscr V(z) = |z|^2 - τ\operatorname{Re}(z_1^2+\ldots+z_d^2)$, where $0\leq τ<1$. Their average density of particles converges to a uniform law on a $2d$-dimensional ellipsoidal region. It is on the hyperellipsoid bounding this region that we find a complementary error function behavior and the Faddeeva plasma kernel. To the best of our knowledge, this is the first instance of the Faddeeva plasma kernel emerging in a higher dimensional model. The results provide evidence for a possible edge universality theorem for determinantal point processes on $\mathbb C^d$.
