AI 中文总结
该研究在二维晶格上提出新渗流模型，通过在网格\(\mathbb{Z}^2\)单元格随机放点，两点距离小于\(R\)则连边，探讨几乎必然出现无限连通分量的半径及另外两个特定临界半径。
AI 中文摘要
我们在二维晶格上提出了一种新的渗流模型，它可视为平面上连续渗流的条件版本。在网格\(\mathbb{Z}^2\)的每个单元格中随机均匀放置一个点，这些点对应图的顶点，若两点距离小于固定半径\(R\)则连接一条边。我们关注几乎必然存在无限连通分量的半径，还研究了该模型几何结构特有的另外两个临界半径：存在点的一种布局使得有无限连通分量的最小半径，以及所有点相互连接的半径。
英文摘要
We present a new percolation model on the two-dimensional lattice, which can be seen as a conditioned version of continuous percolation on the plane. Let us place a point uniformly at random in each cell of the grid $\mathbb{Z}^2$. These points correspond to the vertices of our graph, and we connect two points by an edge if their distance is less than a fixed radius $R$. We are interested in the radius from which there exists almost surely an infinite connected component. We also study two other critical radii specific to the geometry of our model: the smallest radius such that there exists a positioning of the points for which there is an infinite connected component, and the radius from which all points are connected to each other.
