论文标题
比较两个对称集中的受感染顶点的数量（以及其他随机分区）
Comparing the number of infected vertices in two symmetric sets for Bernoulli percolation (and other random partitions)
论文作者
论文摘要
对于给定图$ g =（v，e）$上的bernoulli渗透，我们考虑了一些固定的顶点$ o \ in V $中的群集。我们的目的是比较集合$ v _+$和集合$ v _- $中此群集的顶点的数量，其中$ v _+，v_-- \ subset v $具有相同的大小。直观地，如果$ v _- $距离$ o $远远超过$ v _+$，则应包含群集的更少的顶点。我们证明了在随机统治方面的结果，但前提是v _+$中的$ o \ $ o \ $ v _+，v _- $满足了一些强大的对称条件，并且我们会在此结果的应用中给出$ g $的应用，以防$ g $是层面图，一个分层图，一个层次的图形，一个2D Square Lattice或A HyperCube图。我们的结果仅取决于一般的概率技术和组合结果，因此扩展到相当一般的随机分区，例如由Bernoulli站点渗透或随机群集模型诱导。
For Bernoulli percolation on a given graph $G = (V,E)$ we consider the cluster of some fixed vertex $o \in V$. We aim at comparing the number of vertices of this cluster in the set $V_+$ and in the set $V_-$, where $V_+,V_- \subset V$ have the same size. Intuitively, if $V_-$ is further away from $o$ than $V_+$, it should contain fewer vertices of the cluster. We prove such a result in terms of stochastic domination, provided that $o \in V_+$, and $V_+,V_-$ satisfy some strong symmetry conditions, and we give applications of this result in case $G$ is a bunkbed graph, a layered graph, the 2D square lattice or a hypercube graph. Our result only relies on general probabilistic techniques and a combinatorial result on group actions, and thus extends to fairly general random partitions, e.g. as induced by Bernoulli site percolation or the random cluster model.
