论文标题
在$ \ mathbb {z}^n $上的1个独立渗透的改进范围
Improved bounds for 1-independent percolation on $\mathbb{Z}^n$
论文作者
论文摘要
图$ g $上的1独立债券渗透模型是$ g $的跨度子图上的概率分布，其中所有顶点 - 偶口套件套件$ s_1 $和$ s_2 $，$ s_1 $ s_1 $的边缘状态与S_2 $ $ s_2 $中的Edges的状态无关。如果随机子图具有具有正概率的无限分量，则这种模型被认为会渗透。在2012年，第一作者和Bollobás定义了$ p _ {\ max}（g）$，是那些$ p $的至高无上的$ p $，其中存在一个独立的债券渗透模型上的$ g $，其中每个边缘都会在随机子库中存在至少$ p $，但没有渗透性。 当$ g $是晶格图$ \ mathbb {z}^2 $时，该领域的基本且具有挑战性的问题是确定$ p _ {\ max}（g）$的值。因为$ p _ {\ max}（\ mathbb {z}^n）\ leq p _ {\ max}（\ mathbb {z}^{n-1}）$，也很感兴趣的是，建立$ \ \ \ lim_ {n \ to \ to \ to \ to \ fim iffty} p _ max} $ a {在本文中，我们在此极限上显着改善了最著名的上限，并在$ p _ {\ max}（\ mathbb {z}^2）$上获得更好的上和下限。在证明这些结果时，我们还对超立方体图上的1独立模型的关键概率给出了上限，几乎可以肯定地毫无疑问地包含一个巨大的成分。
A 1-independent bond percolation model on a graph $G$ is a probability distribution on the spanning subgraphs of $G$ in which, for all vertex-disjoint sets of edges $S_1$ and $S_2$, the states of the edges in $S_1$ are independent of the states of the edges in $S_2$. Such a model is said to percolate if the random subgraph has an infinite component with positive probability. In 2012 the first author and Bollobás defined $p_{\max}(G)$ to be the supremum of those $p$ for which there exists a 1-independent bond percolation model on $G$ in which each edge is present in the random subgraph with probability at least $p$ but which does not percolate. A fundamental and challenging problem in this area is to determine the value of $p_{\max}(G)$ when $G$ is the lattice graph $\mathbb{Z}^2$. Since $p_{\max}(\mathbb{Z}^n)\leq p_{\max}(\mathbb{Z}^{n-1})$, it is also of interest to establish the value of $\lim_{n\to\infty} p_{\max}(\mathbb{Z}^n)$. In this paper we significantly improve the best known upper bound on this limit and obtain better upper and lower bounds on $p_{\max}(\mathbb{Z}^2)$. In proving these results, we also give an upper bound on the critical probability for a 1-independent model on the hypercube graph to contain a giant component asymptotically almost surely.
