论文标题
在强钢筋制度中的普通树上温暖的渗透
WARM percolation on a regular tree in the strong reinforcement regime
论文作者
论文摘要
我们在树图上考虑了一类称为温暖的加固过程。这些过程涉及一个参数$α$，该参数控制增强的强度，以及图形顶点索引的泊松过程集合。最近证明，对于任何固定界限图，上面均匀界限的泊松射击速率的任何固定界限图（$α\ gg 1 $ a $ forkimimal tegimimal degimal degimal tem）在非常大的强化方案中，这是“幸存下来”的一组（即经常在流程中强化的）。 目前的论文致力于在相反方向上构造一个示例，即，一组幸存的边缘具有无限连接的组件。也就是说，我们表明，对于每个固定的$α> 1 $，一个人都可以找到一棵常规的生根树和射击速率，这些树从上方均匀地界定，几乎可以肯定地有无限的组件。加入此类示例，我们找到了一个图（带有无限度）的图（对于任何$α> 1 $，几乎肯定都有幸存的边缘的无限连接组件。
We consider a class of reinforcement processes, called WARMs, on tree graphs. These processes involve a parameter $α$ which governs the strength of the reinforcement, and a collection of Poisson processes indexed by the vertices of the graph. It has recently been proved that for any fixed bounded degree graph with Poisson firing rates that are uniformly bounded above, in the very strong reinforcement regime ($α\gg 1$ sufficiently large depending on the maximal degree), the set of edges that "survive" (i.e. that are reinforced infinitely often by the process) has only finite connected components. The present paper is devoted to the construction of an example in the opposite direction, that is, with the set of surviving edges having infinite connected components. Namely, we show that for each fixed $α>1$ one can find a regular rooted tree and firing rates that are uniformly bounded from above, for which there are infinite components almost surely. Joining such examples, we find a graph (with unbounded degrees) on which for any $α>1$ almost surely there are infinite connected components of surviving edges.
