Exact limiting shape for a simplified model of first-passage percolation on the plane
Exact limiting shape for a simplified model of first-passage percolation on the plane
复制标题
平面上第一通道渗流简化模型的精确极限形状
DOI：
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发表时间：
1998
期刊：
影响因子：
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通讯作者：
T. Seppäläinen
中科院分区：
文献类型：
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作者：
T. Seppäläinen
我们推导出以下模型的限制形状的第一次通过键渗流的二维整数晶格：渗透是在这个意义上，允许的路径是非递减的坐标方向。水平键的通过时间服从伯努利分布，而垂直键的通过时间均为确定性常数。为了分析渗流模型，我们将其与一维相互作用粒子系统耦合。这个粒子过程具有非定域动力学，因为任何给定粒子的运动都可能受到远处粒子的影响。我们证明了在这个过程中的一个标记粒子的大数定律，并获得的形状结果的渗流作为一个推论。
We derive the limiting shape for the following model of first-passage bond percolation on the two-dimensional integer lattice: the percolation is directed in the sense that admissible paths are nondecreasing in both coordinate directions. The passage times of horizontal bonds are Bernoulli distributed, while the passage times of vertical bonds are all equal to a deterministic constant. To analyze the percolation model, we couple it with a one-dimensional interacting particle system. This particle process has nonlocal dynamics in the sense that the movement of any given particle can be influenced by far-away particles. We prove a law of large numbers for a tagged particle in this process, and the shape result for the percolation is obtained as a corollary.