边界聚集模型
Border aggregation model
摘要：
从一个顶点子集为{ \it the border}的图开始。从原点释放的粒子在图上执行随机游动，直到它到达边界的紧邻，在该点处它加入该子集，从而将边界增加一个点。然后，一个新的粒子从原点释放出来，这个过程重复，直到原点成为边界本身的一部分。我们感兴趣的是在这个最后时刻释放的粒子的总数$ξ$。我们表明，这个模型涵盖OK畜栏模型以及侵蚀模型，并获得$ξ$的分布和边界的情况下，图是星图，正则树，和$d-$维格。
英文摘要：
Start with a graph with a subset of vertices called {\it the border}. A particle released from the origin performs a random walk on the graph until it comes to the immediate neighbourhood of the border, at which point it joins this subset thus increasing the border by one point. Then a new particle is released from the origin and the process repeats until the origin becomes a part of the border itself. We are interested in the total number $ξ$ of particles to be released by this final moment. We show that this model covers OK Corral model as well as the erosion model, and obtain distributions and bounds for $ξ$ in cases where the graph is star graph, regular tree, and a $d-$dimensional lattice.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
