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An oriented competition model on $$Z_{+}^{2}$$. (English) Zbl 1189.60140
Summary: We consider a two-type oriented competition model on the first quadrant of the two-dimensional integer lattice. Each vertex of the space may contain only one particle of either Red type or Blue type. A vertex flips to the color of a randomly chosen southwest nearest neighbor at exponential rate 2. At time zero there is one Red particle located at $$(1,0)$$ and one Blue particle located at $$(0,1)$$. The main result is a partial shape theorem: Denote by $$R(t)$$ and $$B(t)$$ the red and blue regions at time $$t$$. Then (i) eventually the upper half of the unit square contains no points of $$B(t)/t$$, and the lower half no points of $$R(t)/t$$; and (ii) with positive probability there are angular sectors rooted at $$(1,1)$$ that are eventually either red or blue. The second result is contingent on the uniform curvature of the boundary of the corresponding Richardson shape.

##### MSC:
 60J25 Continuous-time Markov processes on general state spaces 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60J27 Continuous-time Markov processes on discrete state spaces
##### Keywords:
competition; shape theorem; first passage percolation
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