具有瘫痪障碍的随机空间增长
Random spatial growth with paralyzing obstacles
摘要：
我们研究空间增长过程的模型，其中最初有增长的来源（由绿色表示）和增长停止（麻痹）物质的来源（由红色表示）。绿色能源不断扩大，并可能与其他能源合并（不存在“绿色能源间”竞争）。红色物质只要被隔离就保持被动。然而，当一个绿色的簇与红色物质接触时，它立即被后者侵入，停止生长，并开始作为红色物质本身。在我们的主模型空间表示的图形，其中最初的每个顶点是随机的绿色，红色或白色（空），和绿色集群的增长是类似的第一次通过渗流。我们研究的主要问题是模型是否在无限图（例如$d$维立方晶格）上定义良好，以及绿色团簇在瘫痪之前的大小分布可以说什么。我们证明了，如果红色顶点的初始密度是正的，白色顶点的初始密度足够小，模型确实是定义良好的，并且上述分布具有指数尾。事实上，我们相信只要红色的初始密度是正的，这就是真的。这项研究还导致了入侵渗流和临界伯努利渗流之间的关系，这似乎是独立的利益。
英文摘要：
We study models of spatial growth processes where initially there are sources of growth (indicated by the colour green) and sources of a growth-stopping (paralyzing) substance (indicated by red). The green sources expand and may merge with others (there is no `inter-green' competition). The red substance remains passive as long as it is isolated. However, when a green cluster comes in touch with the red substance, it is immediately invaded by the latter, stops growing and starts to act as red substance itself. In our main model space is represented by a graph, of which initially each vertex is randomly green, red or white (vacant), and the growth of the green clusters is similar to that in first-passage percolation. The main issues we investigate are whether the model is well-defined on an infinite graph (e.g. the $d$-dimensional cubic lattice), and what can be said about the distribution of the size of a green cluster just before it is paralyzed. We show that, if the initial density of red vertices is positive, and that of white vertices is sufficiently small, the model is indeed well-defined and the above distribution has an exponential tail. In fact, we believe this to be true whenever the initial density of red is positive. This research also led to a relation between invasion percolation and critical Bernoulli percolation which seems to be of independent interest.
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