An oriented competition model on Z+2
Steven P Lalley, University of Chicago
George Kordzakhia, University of California, Berkeley
Abstract
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.
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