## Electronic Journal of Probability

### Phase transition of the contact process on random regular graphs

#### Abstract

We consider the contact process with infection rate $\lambda$ on a random $(d+1)$-regular graph with $n$ vertices, $G_n$. We study the extinction time $\tau _{G_n}$ (that is, the random amount of time until the infection disappears) as $n$ is taken to infinity. We establish a phase transition depending on whether $\lambda$ is smaller or larger than $\lambda _1(\mathbb{T} ^d)$, the lower critical value for the contact process on the infinite, $(d+1)$-regular tree: if $\lambda < \lambda _1(\mathbb{T} ^d)$, $\tau _{G_n}$ grows logarithmically with $n$, while if $\lambda > \lambda _1(\mathbb{T} ^d)$, it grows exponentially with $n$. This result differs from the situation where, instead of $G_n$, the contact process is considered on the $d$-ary tree of finite height, since in this case, the transition is known to happen instead at the upper critical value for the contact process on $\mathbb{T} ^d$.

#### Article information

Source
Electron. J. Probab., Volume 21 (2016), paper no. 31, 17 pp.

Dates
Accepted: 25 March 2016
First available in Project Euclid: 14 April 2016

https://projecteuclid.org/euclid.ejp/1460652932

Digital Object Identifier
doi:10.1214/16-EJP4476

Mathematical Reviews number (MathSciNet)
MR3492935

Zentralblatt MATH identifier
1342.82104

#### Citation

Mourrat, Jean-Christophe; Valesin, Daniel. Phase transition of the contact process on random regular graphs. Electron. J. Probab. 21 (2016), paper no. 31, 17 pp. doi:10.1214/16-EJP4476. https://projecteuclid.org/euclid.ejp/1460652932

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