Cluster growth in the dynamical Erdős-Rényi process with forest fires
Cluster growth in the dynamical Erdős-Rényi process with forest fires
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森林火灾动态 Erdős-Rényi 过程中的簇增长
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
B. Tóth
中科院分区：
文献类型：
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作者：
Edward Crane;Nic Freeman;B. Tóth
我们研究了拉特和托特的森林火灾模型[EJP，第14卷，第45号论文]中的集群增长。该模型是一个连续时间马尔可夫过程，类似于动态Erdens-Renyi随机图，但增加了所谓的火灾。一个顶点可能在任何时刻着火，当它这样做时，会导致其连接簇内的所有边燃烧，这意味着它们瞬间消失。每个烧焦的边缘可能会在以后重新出现。我们给出了一个精确的描述的过程$C_t$的大小的集群的一个标记的顶点，在限制的顶点数在模型中趋于无穷大。我们证明了$C_t$是一个爆炸分支过程，具有时间非齐次的后代分布，并且在每次爆炸时瞬时返回到1。此外，我们表明，用于分析的Smoluchowski型凝聚方程模型的特征曲线有一个概率解释的过程$C_t$。
We investigate the growth of clusters within the forest fire model of Rath and Toth [EJP, vol 14, paper no 45]. The model is a continuous-time Markov process, similar to the dynamical Erdős-Renyi random graph but with the addition of so-called fires. A vertex may catch fire at any moment and, when it does so, causes all edges within its connected cluster to burn, meaning that they instantaneously disappear. Each burned edge may later reappear. We give a precise description of the process $C_t$ of the size of the cluster of a tagged vertex, in the limit as the number of vertices in the model tends to infinity. We show that $C_t$ is an explosive branching process with a time-inhomogeneous offspring distribution and instantaneous return to 1 on each explosion. Additionally, we show that the characteristic curves used to analyse the Smoluchowski-type coagulation equations associated to the model have a probabilistic interpretation in terms of the process $C_t$.