论文标题
关于独立比对渗透的相变的注释
A note on the phase transition for independent alignment percolation
论文作者
论文摘要
我们研究了Beaton，Grimmett和Holmes引入的$ \ Mathbb {Z}^D $上的独立对齐渗透模型[arxiv：1908.07203]。它是定义如下的随机相交线段的模型。首先，$ \ mathbb {z}^d $的网站被独立声明，占据了概率$ p $，否则就空置了。以占领顶点的配置为条件，考虑与坐标轴平行的所有线段的集合，坐标轴的极端是占据的顶点，并且不会横穿任何其他被占据的顶点。以概率$λ$开放，独立声明此集合中的细分市场，否则就关闭了。在$ \ mathbb {z}^d $中，所有位于开放片段的边缘也被宣布为开放式的债券渗透模型。 We show that for any $d \geq 2$ and $p \in (0,1]$ the critical value for $λ$ satisfies $λ_c(p)<1$ completing the proof that the phase transition is non-trivial over the whole interval $(0,1]$. We also show that the critical curve $p \mapsto λ_c(p)$ is continuous at $p=1$, answering a question posed by the authors in [Arxiv：1908.07203]。
We study the independent alignment percolation model on $\mathbb{Z}^d$ introduced by Beaton, Grimmett and Holmes [arXiv:1908.07203]. It is a model for random intersecting line segments defined as follows. First the sites of $\mathbb{Z}^d$ are independently declared occupied with probability $p$ and vacant otherwise. Conditional on the configuration of occupied vertices, consider the set of all line segments that are parallel to the coordinate axis, whose extremes are occupied vertices and that do not traverse any other occupied vertex. Declare independently the segments on this set open with probability $λ$ and closed otherwise. All the edges that lie on open segments are also declared open giving rise to a bond percolation model in $\mathbb{Z}^d$. We show that for any $d \geq 2$ and $p \in (0,1]$ the critical value for $λ$ satisfies $λ_c(p)<1$ completing the proof that the phase transition is non-trivial over the whole interval $(0,1]$. We also show that the critical curve $p \mapsto λ_c(p)$ is continuous at $p=1$, answering a question posed by the authors in [arXiv:1908.07203].
