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Competition in growth and urns
Stockholm University, Faculty of Science, Department of Mathematics.
Number of Authors: 42019 (English)In: Random structures & algorithms (Print), ISSN 1042-9832, E-ISSN 1098-2418, Vol. 54, no 2, p. 211-227Article in journal (Refereed) Published
Abstract [en]

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 infinity, 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.

Place, publisher, year, edition, pages
2019. Vol. 54, no 2, p. 211-227
Keywords [en]
bootstrap percolation, branching processes, competing growth, urn models
National Category
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-166676DOI: 10.1002/rsa.20779ISI: 000458197400001OAI: oai:DiVA.org:su-166676DiVA, id: diva2:1293449
Available from: 2019-03-04 Created: 2019-03-04 Last updated: 2019-03-04Bibliographically approved

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