随机Voronoi镶嵌上的贪婪多项式和首次通过时间
Greedy Polyominoes and first-passage times on random Voronoi tilings
摘要：
设N分布为R^d上的泊松随机集，其强度与勒贝格测度相当。考虑R^d的Voronoi平铺，（C_v）_{v \in N}，其中C_v由R^d中的点x组成，这些点x比N中的任何其他v'更接近v。一个大小为n的多项式P是n个瓦片的连通并集（在通常的R^d拓扑意义上），我们用Pi_n表示所有包含原点的大小为n的多项式P的集合。假设Voronoi图块C_v的权重由F（C_v）给出，其中F是Voronoi图块上的非负泛函。本文主要研究了当F（Cv）为Cv的面数时，函数F在Pi_n中的最大权的尾部行为，并将所得结果应用于自避免路径、首次通过渗流模型和对偶图上的刺数，即Delaunay三角剖分.作为主要的应用程序，我们表明，第一通道渗流最多线性方差。
英文摘要：
Let N be distributed as a Poisson random set on R^d with intensity comparable to the Lebesgue measure. Consider the Voronoi tiling of R^d, (C_v)_{v\in N}, where C_v is composed by points x in R^d that are closer to v than to any other v' in N. A polyomino P of size n is a connected union (in the usual R^d topological sense) of n tiles, and we denote by Pi_n the collection of all polyominos P of size n containing the origin. Assume that the weight of a Voronoi tile C_v is given by F(C_v), where F is a nonnegative functional on Voronoi tiles. In this paper we investigate the tail behavior of the maximal weight among polyominoes in Pi_n for some functionals F, mainly when F(C_v) is the number of faces of C_v. Next we apply our results to study self-avoiding paths, first-passage percolation models and the stabbing number on the dual graph, named the Delaunay triangulation. As the main application we show that first passage percolation has at most linear variance.
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