A Geometric Perspective on First-Passage Competition

Abstract

We study the macroscopic geometry of first-passage competition on the integer lattice <bold>Z</bold><super>d</super>, with a particular interest in describing the behavior when one species initially occupies the exterior of a cone. First-passage competition is a stochastic process modeling two infections spreading outward from initially occupied disjoint subsets of <bold>Z</bold><super>d</super>. Each infecting species transmits its infection at random times from previously infected sites to neighboring uninfected sites. The infection times are governed by species-specific probability distributions, and every vertex of <bold>Z</bold><super>d</super> remains permanently infected by whichever species infects it first. We introduce a new, simple construction of first-passage competition that works for an arbitrary pair of disjoint starting sets in <bold>Z</bold><super>d</super>, and we analogously define a deterministic first-passage competition process in the Euclidean space <bold>R</bold><super>d</super>, providing a formal definition for a model of crystal growth that has previously been studied computationally. We then prove large deviations estimates for the random <bold>Z</bold><super>d</super>-process, showing that on large scales it is well-approximated by the deterministic <bold>R</bold><super>d</super>-process, with high probability. Analyzing the geometry of the deterministic process allows us to identify critical phenomena in the random process when one of the two species initially occupies the entire exterior of a cone and the other species initially occupies a single interior site. Our results generalize those in a 2007 paper of Deijfen and Haggstrom, who considered the case where the cone is a half-space. Moreover, we use our results about competition in cones to strengthen a 2000 result of Haggstrom and Pemantle about competition from finite starting configurations.