## Electronic Communications in Probability

### An oriented competition model on $Z_+^2$

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

#### Article information

Source
Electron. Commun. Probab., Volume 13 (2008), paper no. 52, 548-561.

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

https://projecteuclid.org/euclid.ecp/1465233479

Digital Object Identifier
doi:10.1214/ECP.v13-1422

Mathematical Reviews number (MathSciNet)
MR2453548

Zentralblatt MATH identifier
1189.60140

Rights
Lalley, Steven; Kordzakhia, George. An oriented competition model on $Z_+^2$. Electron. Commun. Probab. 13 (2008), paper no. 52, 548--561. doi:10.1214/ECP.v13-1422. https://projecteuclid.org/euclid.ecp/1465233479