Electronic Communications in Probability
- Electron. Commun. Probab.
- Volume 13 (2008), paper no. 52, 548-561.
An oriented competition model on $Z_+^2$
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.
Electron. Commun. Probab., Volume 13 (2008), paper no. 52, 548-561.
Accepted: 18 October 2008
First available in Project Euclid: 6 June 2016
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
This work is licensed under aCreative Commons Attribution 3.0 License.
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