Bernoulli percolation on the Random Geometric Graph
Bernoulli percolation on the Random Geometric Graph
复制标题
随机几何图上的伯努利渗滤
DOI：
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发表时间：
2022
期刊：
影响因子：
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通讯作者：
Bruno Schapira
中科院分区：
文献类型：
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作者：
Lyuben Lichev;B. Lodewijks;D. Mitsche;Bruno Schapira
设λ > 0，p ∈ [0，1]，R 2中的Poisson点过程Po（λ）强度为λ，考虑顶点集为Po（λ）的随机图G = G（λ，p），其中每对距离不超过1的顶点独立于其它顶点对以概率p形成一条边.我们的研究是基于标准随机几何图G（λ，1）上键渗流的局部性问题。我们证明了在适当的意义下，对于一类局部收敛于G（λ，1）的图，其临界渗流阈值收敛于G（λ，1）的临界渗流阈值.证明是基于一个有限体积的标准。在这个方向上，我们加强最近的结果彭罗斯表明，最大的组件的大小，重新缩放的n，几乎肯定收敛到一个常数，并提供尖锐的边界的大小第二大组件。
Given λ > 0 , p ∈ [0 , 1] and a Poisson Point Process Po( λ ) in R 2 with intensity λ , we consider the random graph G = G ( λ, p ) with vertex set Po( λ ) in which every pair of vertices at distance at most 1 forms an edge with probability p , independently of other pairs. Our study is motivated by the question of locality for bond percolation on the standard random geometric graph G ( λ, 1) . We show that for a large class of graphs converging locally to G ( λ, 1) in a suitable sense, the corresponding critical percolation thresholds converge to the one of G ( λ, 1) . The proof is based on a finite volume criterion. In this direction, we strengthen recent results of Penrose by showing that the size of the largest component, rescaled by n , converges almost surely to a constant, and providing sharp bounds for the size of the second-largest component.
影响因子：
0.5
作者：
Penrose M
通讯作者：
Penrose M