Richardson型竞争空间增长模型的共存性
Coexistence for Richardson type competing spatial growth models
摘要：
我们研究了一个大家庭的竞争空间增长模型。在这些中，Z^d中的顶点可以呈现三种可能的状态{0，1，2}。处于状态1和2的顶点永远保持其状态，而与处于状态1（或状态2）的顶点相邻的处于状态0的顶点可以切换到状态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的顶点被认为是未感染的。这样，这些模型是理查森模型的变体。我们从一个处于状态1的顶点和一个处于状态2的顶点开始模型。我们表明，正概率状态1达到了无穷多个顶点，状态2也达到了无穷多个顶点。这推广了已有的结果，并证明了Haggstrom和Pemantle的一个猜想.关键工具是将遍历定理应用于稳态第一通道渗流。
英文摘要：
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.
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