Electronic Communications in Probability

An oriented competition model on $Z_+^2$

Steven Lalley and George Kordzakhia

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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

Permanent link to this document
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

Subjects
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]

Keywords
competition shape theorem first passage percolation

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

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


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