Competition in growth and urns

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.bootstrap percolation, branching processes, competing growth, urn models A model of competition for space between two or more growing entities was introduced in the context of first-passage percolation on Z 2 by Häggström and Pemantle [22]. Beyond the mere beauty of the Random Struct Alg. 2018;00:1-17.wileyonlinelibrary.com/journal/rsa

To fixate or not to fixate in two-type annihilating branching random walks

“…We now, finally, turn to the two-colour competition process. We introduce a conservative version of the process, previously explored in [2]. In this process, red and blue balls branch and get offspring as in the competion process described in the introduction, but when a red and a blue ball meet, instead of annihilating, the two balls merge to form a purple ball.…”

“…1 Assuming that the offspring distribution is irreducible means no loss of generality, since we otherwise could consider the process on the subgroup G of Z d generated by the support of Φ; note that G ∼ = Z d 1 for some d1 d and that the process on Z d then decomposes into independent copies of the process on G, supported on different translates (cosets) of G. We ignore the trivial case when the support of Φ is {0}; then the urns are independent continuous-time branching processes, each with a fixed colour. 2 The times the different urns fixate are random and different; we do not claim that there is a single time when all urn have the same colour. (Indeed, since the system is infinite, we cannot expect this.…”

“…In the case that G is connected and finite, the authors together with Morris [2] have proved that the system of urns will eventually almost surely contain balls of only one colour. In the current paper we examine the process on the d-dimensional integer lattice Z d , for d 1, evolving from an initial configuration with a ball at each site, which independently from one another are coloured red with probability p and blue otherwise.…”

“…As before, all children has the same colour as their parent, and if one or several balls are positioned in an urn with balls of opposite colour, then they annihilate one for one until all remaining balls are of the same colour. The nearest-neighbour rule (from [2]) described above thus corresponds to Φ being the degenerate measure supported on the configuration consisting of one ball at each of the 2d neighbours to the origin. We shall at later occasions in the paper have more to say about the construction of the process as the need arises.…”

“…We then return to the two-colour competition process, starting from a p-random Bernoulli colouring, which we describe by the vector Z(t) = (Z x (t)) x∈Z d . Although our main interest lies in the case of a random initial configuration, we will in the proofs consider various versions, and we thus allow an arbitrary initial configuration We describe in Section 4 a coupling, previously employed in [2], that enables us to ignore annihilations and instead study a pair of (dependent) monochromatic processes, to which we can apply results from previous sections. Theorem 1.2(i) then is as an easy consequence of Theorem 1.5.…”