Competition in growth and urns. Issue 2 (15th April 2018)

• Record Type: Journal Article
• Authors: Ahlberg, Daniel; Griffiths, Simon; Janson, Svante; Morris, Robert
• Is Part Of: Random structures & algorithms. Volume 54:Issue 2(2019)
• Page Start: 211
• Page End: 227
• Publication Date: 2018-04-15
• Subjects: bootstrap percolation -- branching processes -- competing growth -- urn models; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20779
• Languages: English
• ISSNs: 1042-9832

Abstract

Abstract: We 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 ℤ 2, 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.