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2008 An oriented competition model on $Z_+^2$
Steven Lalley, George Kordzakhia
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Electron. Commun. Probab. 13: 548-561 (2008). DOI: 10.1214/ECP.v13-1422


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 $$. The main result is a partial shape theorem: Denote by $R (t)$ and $B (t)$ the red and blue regions at time $t$. Then (i) eventually the upper half of the unit square contains no points of $B (t)/t$, and the lower half no points of $R (t)/t$; and (ii) with positive probability there are angular sectors rooted at $$ that are eventually either red or blue. The second result is contingent on the uniform curvature of the boundary of the corresponding Richardson shape.


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Steven Lalley. George Kordzakhia. "An oriented competition model on $Z_+^2$." Electron. Commun. Probab. 13 548 - 561, 2008.


Accepted: 18 October 2008; Published: 2008
First available in Project Euclid: 6 June 2016

zbMATH: 1189.60140
MathSciNet: MR2453548
Digital Object Identifier: 10.1214/ECP.v13-1422

Primary: 60J25
Secondary: 60K35

Keywords: competition , first passage percolation , shape theorem

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