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Abstract:
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In threshold growth models, we study a deterministic process that iteratively enlarges a set of occupied sites by adjoining points with at least r occupied neighbours. In the polluted threshold growth model on Z^2, the vertices are independently declared occupied with probability p, closed with probability q and empty otherwise. For any x in Z^2, first its neighbourhood structure is defined appropriately. At any integer valued time point t, an empty vertex gets occupied if at least r of its neighbours are occupied at time t-1. Vertices which are already occupied or closed do not change state. Our objective is to study the final density of occupied sites as p, q approach 0 and to see if there is any critical scaling relation between p and q which affects the probability that any vertex is eventually occupied.
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