Coexistence for Richardson type competing spatial growth models
Coexistence for Richardson type competing spatial growth models
复制标题
理查森型竞争空间增长模型的共存
DOI：
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
C. Hoffman
中科院分区：
文献类型：
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作者：
C. Hoffman
我们研究了一大类相互竞争的空间增长模型。在这种情况下，Z^d中的顶点可以有三种可能的状态{0,1,2}。处于状态1和状态2的顶点永远保持其状态，而处于状态0的顶点与处于状态1（或状态2）的顶点相邻，可以切换到状态1（或状态2）。我们认为处于状态1和2的顶点感染了两种感染中的一种，而处于状态0的顶点被认为未感染。这样看来，这些模型都是理查森模型的变体。我们以状态1的单个顶点和状态2的单个顶点开始模型。我们证明了以正概率状态1到达无限个顶点状态2也到达无限个顶点。这扩展了结果并证明了Haggstrom和Pemantle的一个猜想。关键的工具是将遍历定理应用于稳态第一通道渗流。
We study a large family of competing spatial growth models. In these the vertices in Z^d can take on three possible states {0,1,2}. Vertices in states 1 and 2 remain in their states forever, while vertices in state 0 which are adjacent to a vertex in state 1 (or state 2) can switch to state 1 (or state 2). We think of the vertices in states 1 and 2 as infected with one of two infections while the vertices in state 0 are considered uninfected. In this way these models are variants of the Richardson model. We start the models with a single vertex in state 1 and a single vertex is in state 2. We show that with positive probability state 1 reaches an infinite number of vertices and state 2 also reaches an infinite number of vertices. This extends results and proves a conjecture of Haggstrom and Pemantle. The key tool is applying the ergodic theorem to stationary first passage percolation.