A few years ago two of us introduced, motivated by the study of certain forest-fireprocesses, the self-destructive percolation model (abbreviated as sdp model). A typical configuration for the sdp model with parameters p and delta is generated in three steps: First we generate a typical configuration for the ordinary percolation model with parameter p. Next, we make all sites in the infinite occupied cluster vacant. Finally, each site that was already vacant in the beginning or made vacant by the above action, becomes occupied with probability delta (independent of the other sites).
Let theta(p, delta) be the probability that some specified vertex belongs, in the final configuration, to an infinite occupied cluster. In our earlier paper we stated the conjecture that, for the square lattice and other planar lattices, the function theta has a discontinuity at points of the form (p_c, delta), with delta sufficiently small. We also showed remarkable consequences for the forest-fire models.
The conjecture naturally raises the question whether the function theta is continuous outside some region of the above mentioned form. We prove that this is indeed the case. An important ingredient in our proof is a (somewhat stronger form of a) recent ingenious RSW-like percolation result of Bollobás and Riordan.
Bernoulli and self-destructive percolation on non-amenable graphs
本文研究了非顺从图上无限渗流簇的一些性质。特别地，我们研究了无限渗流集团的补的分解性质。本文采用基于质量迁移的方法，证明了对于一大类不服从图，通过去除包含在无限渗流簇中的每一个点而得到的图具有临界渗流阈值，当p接近pc时，该临界渗流阈值可以任意地接近于原始图的临界阈值.与之密切相关的是由J. van den Berg和R. Brouwer，我们证明了一个无限的集群出现任何小的加强。
Linear Lower Bounds for $δ_c(p)$ for a Class of 2D Self-Destructive Percolation Models
自破坏渗流模型定义如下：考虑参数为$p > p_c$的渗流。移除无限占用群集。最后，给每个顶点（或者，对于键渗透，每个边）一个额外的机会，使其在这个阶段是空的，$δ$被占据。令$δ_c(p)$为$δ$的极小值，需要在最终配置中获得无限占用的簇。这个模型是几年前由van den Berg和Brouwer提出的。他们表明，对于正方形网格（以及其他一些满足特殊技术条件的二维网格）上的站点模型，$δ_c(p)\geq\frac{(p-p_c)}{p}$。特别地，$δ_c(p)$在$p-p_c$中至少是线性的。虽然范登伯格和布劳威尔使用的参数看起来相当严格，我们表明，他们可以适当修改，以获得类似的线性下界$δ_c(p)$（与$p$附近$p_c$）的一个更大的类的二维晶格，包括键渗流的正方形和三角形晶格，和网站渗流的星形晶格（或匹配晶格）的正方形晶格。
