Greedy Polyominoes and first-passage times on random Voronoi tilings
Abstract
Let N be distributed as a Poisson random set on Rd with intensity comparable to the Lebesgue measure. Consider the Voronoi tiling of Rd, (Cv)v∈ N, where Cv is composed by points x in Rd 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 Rd topological sense) of n tiles, and we denote by Pin the collection of all polyominos P of size n containing the origin. Assume that the weight of a Voronoi tile Cv is given by F(Cv), where F is a nonnegative functional on Voronoi tiles. In this paper we investigate the tail behavior of the maximal weight among polyominoes in Pin for some functionals F, mainly when F(Cv) is the number of faces of Cv. 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.