随机帽覆盖球的强阈值

用四个随机圆帽覆盖球面

Covering a Sphere with Four Random Circular Caps

设p（w）表示四个随机圆帽覆盖单位球面S^2的概率，圆帽的角半径为70度<w<90度。p（w）的精确表达式是未知的。当w> 84 deg时，我们给出了p（w）的非平凡下界;当w<84 deg时，对不等式p（w）>=0的改进是不可行的。一个对偶问题，涉及随机内接井心四面体也检查。

球面上随机交图的边数

Number of Edges in Random Intersection Graph on Surface of a Sphere

在这篇文章中，我们认为“$N$”球冠面积$4πp$均匀分布在一个单位球的表面上。我们研究了由这些帽构造的随机交图$G_N$。证明了对于$p = \frac{c}{N^{\al}},\:c >0$和$\al >2,$，图$G_N$的边数服从Poisson分布.我们还得到了关于$G_N$中孤立点个数的强律结果：对于$p = \frac{c}{N^{\al}},\:c >0$对于$\al < 1,$，几乎必然在$G_N$中没有孤立点，即，在$G_N$中至少存在$N/2$边并且对于$\al >3,$，$G_N$中的每个顶点是孤立的即，边集合$\cE_N.$中没有边

A Quantitative Central Limit Theorem for the Excursion Area of Random Spherical Harmonics over Subdomains of $\mathbb{S}^2$

In recent years, considerable interest has been drawn by the analysis of geometric functionals for the excursion sets of random eigenfunctions on the unit sphere (spherical harmonics). In this paper, we extend those results to proper subsets of the sphere $\mathbb{S}^2$, i.e., spherical caps, focussing in particular on the excursion area. Precisely, we show that the asymptotic behavior of the excursion area is dominated by the so-called second-order chaos component, and we exploit this result to establish a Quantitative Central Limit Theorem, in the high energy limit. These results generalize …

球面上的随机点集--洞半径、覆盖和分离

Random Point Sets on the Sphere --- Hole Radii, Covering, and Separation

研究了单位球面$\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$上独立均匀分布的$N$随机点关于表面积测度的几何性质，并给出了几个相关的证明.特别是，我们推导出渐近（$N \to \infty$）的预期时刻的半径与小平面的凸包的$N$随机点$\mathbb{S}^{d}$的球冠。我们提供了这些球冠的缩放半径的渐近分布和这些半径（覆盖半径）的最大值的期望值的示意图。数值证据包括支持这些acquitures。此外，利用Cai等人的成对角度极值定律，我们推导出$\mathbb{S}^{d}$上随机点的期望分离的精确渐近性。

Tightness for the Cover Time of the two dimensional sphere

Let $C^*_{ε,S^2}$ denote the cover time of the two dimensional sphere by a Wiener sausage of radius $ε$. We prove that $$\sqrt{C^{*}_{ε,S^2} } -\sqrt{\frac{2A_{S^2}}π}(\log ε^{-1}-\frac14\log\log ε^{-1})$$ is tight, where $A_{S^2}=4π$ denotes the Riemannian area of $S^2$.

高维中随机部分球面覆盖的中心极限定理

Central Limit Theorem for Random Partial Sphere Coverings in High Dimensions

研究高维单位球面上随机部分覆盖模型，通过建立中心极限定理，证明随机覆盖体积的波动渐近正态，并给出收敛速率的定量界。