## Electronic Communications in Probability

### Coexistence in chase-escape

#### Abstract

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.

#### Article information

Source
Electron. Commun. Probab., Volume 25 (2020), paper no. 22, 14 pp.

Dates
Accepted: 25 February 2020
First available in Project Euclid: 4 March 2020

https://projecteuclid.org/euclid.ecp/1583290992

Digital Object Identifier
doi:10.1214/20-ECP302

#### Citation

Durrett, Rick; Junge, Matthew; Tang, Si. Coexistence in chase-escape. Electron. Commun. Probab. 25 (2020), paper no. 22, 14 pp. doi:10.1214/20-ECP302. https://projecteuclid.org/euclid.ecp/1583290992

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