论文标题
磁盘计数统计数据靠近随机正常矩阵的硬边缘：多组分制度
Disk counting statistics near hard edges of random normal matrices: the multi-component regime
论文作者
论文摘要
我们考虑了一个二维点过程，该过程通过硬墙将点分为两个不相交的组件，并研究相应的磁盘计数统计量的多变量力矩生成函数。 We investigate the ``hard edge regime" where all disk boundaries are a distance of order $\frac{1}{n}$ away from the hard wall, where $n$ is the number of points. We prove that as $n \to + \infty$, the asymptotics of the moment generating function are of the form \begin{align*} & \exp \bigg(C_{1}n + c_ {2} \ ln n + c_ {3} + \ Mathcal {f} _ {n} + \ frac {c_ {4}} {\ sqrt {\ sqrt {n}}}}} + \ \ \ \ \ \ \ m rathcal {o} \ end {Align*}，我们确定$ c_ {1}，\ dots，c_ {4} $明确说明振荡术语$ \ nathcal {f} _ {n} $是$ 1 $的$ 1 $。一个组件中点数的渐近波动是$ 1 $，并且由振荡性离散的高斯给出，此随机变量的差异符合Weiersstrass $ \ wp $ - 功能。
We consider a two-dimensional point process whose points are separated into two disjoint components by a hard wall, and study the multivariate moment generating function of the corresponding disk counting statistics. We investigate the ``hard edge regime" where all disk boundaries are a distance of order $\frac{1}{n}$ away from the hard wall, where $n$ is the number of points. We prove that as $n \to + \infty$, the asymptotics of the moment generating function are of the form \begin{align*} & \exp \bigg(C_{1}n + C_{2}\ln n + C_{3} + \mathcal{F}_{n} + \frac{C_{4}}{\sqrt{n}} + \mathcal{O}(n^{-\frac{3}{5}})\bigg), \end{align*} and we determine the constants $C_{1},\dots,C_{4}$ explicitly. The oscillatory term $\mathcal{F}_{n}$ is of order $1$ and is given in terms of the Jacobi theta function. Our theorems allow us to derive various precise results on the disk counting function. For example, we prove that the asymptotic fluctuations of the number of points in one component are of order $1$ and are given by an oscillatory discrete Gaussian. Furthermore, the variance of this random variable enjoys asymptotics described by the Weierstrass $\wp$-function.
