Nucleation and growth in two dimensions

Bollobás, Béla; Griffiths, Simon; Morris, Robert; Rolla, Leonardo T.; Smith, Paul

doi:10.1002/rsa.20888

Cited by 3 publications

(3 citation statements)

References 29 publications

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“…We remark that the case λ = 1 corresponds to a close relative of first-passage percolation, in which sites instead of bonds are assigned random weights, known as the Eden model [20]. A more complicated 'nucleation and growth' model, in which sites can become infected despite having no infected neighbours, was studied in [16,12,10], and applied to the Ising model in [15,13].…”

Section: Competition In Growthmentioning

confidence: 99%

Competition in growth and urns

Ahlberg¹,

Griffiths²,

Janson³

et al. 2018

Random Struct Algorithms

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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

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Section: Competition In Growthmentioning

confidence: 99%

Competition in growth and urns

Ahlberg¹,

Griffiths²,

Janson³

et al. 2018

Random Struct Algorithms

Self Cite

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show abstract

“…In [2] the authors were concerned with a variant of the Eden model, in which vacant sites with one occupied neighbour become occupied at rate 1, whereas vacant sites with at least two occupied neighbours become occupied immediately. The Eden model and the variant of the Eden model from [2] constitute two extreme point of a larger family of nucleation and growth processes introduced in [19], and further studied in [9,12,13]. The immediate occupation of sites with at least two occupied neighbours results in a bootstraping effect, similar to that of bootstrap percolation and related automata.…”

Section: Competing Growth On Zmentioning

confidence: 99%

“…Or, as long as there is at least one particle present at time zero, there is positive probability of a (series of nucleations leading to a) nucleation at some node k at which v is nonzero, before t = 1. By (9), this has a positive contribution to the quadratic variation, which by (8) gives…”

Section: Martingale Analysismentioning

confidence: 99%

Multi-colour competition with reinforcement

Ahlberg¹,

Fransson²

2022

Preprint

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We study a system of interacting urns where balls of different colour/type compete for their survival, and annihilate upon contact. For competition between two types, the underlying graph (finite and connected), determining the interaction between the urns, is known to be irrelevant for the possibility of coexistence, whereas for K ≥ 3 types the structure of the graph does affect the possibility of coexistence. We show that when the underlying graph is a cycle, competition between K ≥ 3 types almost surely has a single survivor, thus establishing a conjecture of Griffiths, Janson, Morris and the first author. Along the way, we give a detailed description of an auto-annihilative process on the cycle, which can be perceived as an expression of the geometry of a Möbius strip in a discrete setting.We shall in this paper be concerned with a model of interacting urns where balls of different type annihilate upon contact. Previous work [2] have revealed that the spatial component, in this setting, has an effect on the eventual fate of the process, as we describe further below. The annihilative feature of the process has in the physics literature been

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Multi-colour competition with reinforcement

Ahlberg,

Fransson

2024

Ann. Inst. H. Poincaré Probab. Statist.

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Nucleation and growth in two dimensions

Cited by 3 publications

References 29 publications

Competition in growth and urns

Competition in growth and urns

Multi-colour competition with reinforcement

Multi-colour competition with reinforcement

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