Open Access
2020 Coexistence in chase-escape
Rick Durrett, Matthew Junge, Si Tang
Electron. Commun. Probab. 25: 1-14 (2020). DOI: 10.1214/20-ECP302


We study a competitive stochastic growth model called chase-escape in which red particles spread to adjacent uncolored sites and blue particles only to adjacent red sites. Red particles are killed when blue occupies the same site. If blue has rate-1 passage times and red rate-$\lambda $, a phase transition occurs for the probability red escapes to infinity on $\mathbb{Z} ^{d}$, $d$-ary trees, and the ladder graph $\mathbb{Z} \times \{0,1\}$. The result on the tree was known, but we provide a new, simpler calculation of the critical value, and observe that it is a lower bound for a variety of graphs. We conclude by showing that red can be stochastically slower than blue, but still escape with positive probability for large enough $d$ on oriented $\mathbb{Z} ^{d}$ with passage times that resemble Bernoulli bond percolation.


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Rick Durrett. Matthew Junge. Si Tang. "Coexistence in chase-escape." Electron. Commun. Probab. 25 1 - 14, 2020.


Received: 5 June 2019; Accepted: 25 February 2020; Published: 2020
First available in Project Euclid: 4 March 2020

zbMATH: 1434.60285
MathSciNet: MR4089729
Digital Object Identifier: 10.1214/20-ECP302

Primary: 60K35

Keywords: Growth model , Particle system , phase transition

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