Open Access
2020 The symbiotic contact process
Rick Durrett, Dong Yao
Electron. J. Probab. 25: 1-21 (2020). DOI: 10.1214/19-EJP402


We consider a contact process on $\mathbb{Z} ^{d}$ with two species that interact in a symbiotic manner. Each site can either be vacant or occupied by individuals of species $A$ and/or $B$. Multiple occupancy by the same species at a single site is prohibited. The name symbiotic comes from the fact that if only one species is present at a site then that particle dies with rate 1 but if both species are present then the death rate is reduced to $\mu \le 1$ for each particle at that site. We show the critical birth rate $\lambda _{c}(\mu )$ for weak survival is of order $\sqrt{\mu } $ as $\mu \to 0$. Mean-field calculations predict that when $\mu < 1/2$ there is a discontinuous transition as $\lambda $ is varied. In contrast, we show that, in any dimension, the phase transition is continuous. To be fair to the physicists that introduced the model, [27], the authors say that the symbiotic contact process is in the directed percolation universality class and hence has a continuous transition. However, a 2018 paper, [30], asserts that the transition is discontinuous above the upper critical dimension, which is 4 for oriented percolation.


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Rick Durrett. Dong Yao. "The symbiotic contact process." Electron. J. Probab. 25 1 - 21, 2020.


Received: 4 April 2019; Accepted: 8 December 2019; Published: 2020
First available in Project Euclid: 16 January 2020

zbMATH: 1441.60077
MathSciNet: MR4059182
Digital Object Identifier: 10.1214/19-EJP402

Primary: 60K35

Keywords: block construction , symbiotic contact process

Vol.25 • 2020
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