论文标题
在渗透的随机几何图中的第一大部分和第二大组件上
On the first and second largest components in the percolated Random Geometric Graph
论文作者
论文摘要
渗透的随机几何图$ g_n（λ，p）$具有由平方$ [0，\ sqrt {n}]^2 $中的泊松点进程给出的顶点，并且最多距离的每对距离的每个顶点都独立形成一个边缘，带有概率$ p $。对于固定的$ P $，Penrose证明存在一个关键强度$λ_c=λ_c（p）$，用于存在$ g_n（λ，p）$中的巨型组件。我们的主要结果表明，对于$λ>λ_c$，第二大组件的大小为A.A.S.订单$（\ log n）^2 $的订单。此外，我们证明，最大的组件的大小几乎可以肯定地收敛到恒定，从而增强了彭罗斯的结果。 我们通过显示与泊松强度相关的渗透阈值与$ g（λ，p）$相关的渗透阈值（这是$ g_n（λ，p）$的无限体积版本）之间的一定二元性结果。此外，我们证明，对于以合适的意义收敛到$ g（λ，1）$的大量图，相应的关键渗透阈值也收敛于$ g（λ，1）$的$ g（λ，1）$。
The percolated random geometric graph $G_n(λ, p)$ has vertex set given by a Poisson Point Process in the square $[0,\sqrt{n}]^2$, and every pair of vertices at distance at most 1 independently forms an edge with probability $p$. For a fixed $p$, Penrose proved that there is a critical intensity $λ_c = λ_c(p)$ for the existence of a giant component in $G_n(λ, p)$. Our main result shows that for $λ> λ_c$, the size of the second-largest component is a.a.s. of order $(\log n)^2$. Moreover, we prove that the size of the largest component rescaled by $n$ converges almost surely to a constant, thereby strengthening results of Penrose. We complement our study by showing a certain duality result between percolation thresholds associated to the Poisson intensity and the bond percolation of $G(λ, p)$ (which is the infinite volume version of $G_n(λ,p)$). Moreover, we prove that for a large class of graphs converging in a suitable sense to $G(λ, 1)$, the corresponding critical percolation thresholds converge as well to the ones of $G(λ,1)$.
