Competition in growth and urns

Abstract

AbstractWe study survival among two competing types in two settings: a planar growth model related to two‐neighbor bootstrap percolation, and a system of urns with graph‐based interactions. In the planar growth model, uncolored sites are given a color at rate 0, 1 or , depending on whether they have zero, one, or at least two neighbors of that color. In the urn scheme, each vertex of a graph G has an associated urn containing some number of either blue or red balls (but not both). At each time step, a ball is chosen uniformly at random from all those currently present in the system, a ball of the same color is added to each neighboring urn, and balls in the same urn but of different colors annihilate on a one‐for‐one basis. We show that, for every connected graph G and every initial configuration, only one color survives almost surely. As a corollary, we deduce that in the two‐type growth model on , one of the colors only infects a finite number of sites with probability one. We also discuss generalizations to higher dimensions and multi‐type processes, and list a number of open problems and conjectures.