# The Symbiotic Contact Process

## TL;DR

This paper investigates a two-species symbiotic contact process on $Z^d$, revealing that the phase transition is continuous in all dimensions despite mean-field predictions of discontinuity for certain parameters.

## Contribution

It demonstrates that the symbiotic contact process exhibits a continuous phase transition in any dimension, challenging mean-field predictions of discontinuity for some parameter ranges.

## Key findings

- Critical birth rate $\lambda_c(\mu)$ scales as $\sqrt{\mu}$ for small $\mu$
- Phase transition remains continuous in all dimensions
- Contrasts mean-field predictions of discontinuity for $\mu<1/2$

## Abstract

We consider a contact process on $Z^d$ 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 physicists the paper that introduced the model, 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 asserts that the transition is discontinuous above the upper critical dimension, which is 4 for oriented percolation.