## The Annals of Probability

- Ann. Probab.
- Volume 20, Number 2 (1992), 696-723.

### The Survival of One-Dimensional Contact Processes in Random Environments

#### Abstract

Consider the inhomogeneous contact process on $Z^1$ with recovery rate $\delta(k)$ at site $k$ and infection rates $\lambda(k)$ and $\rho(k)$ at site $k$ due to the presence of infected neighbors at $k - 1$ and $k + 1$ respectively. A special case of the main result in this paper is the following: Suppose that the environment is chosen in such a way that the $\delta(k)$'s, $\lambda(k)$'s and $\rho(k)$'s are all mutually independent, with the $\delta(k)$'s having a common distribution, and the $\lambda(k)$'s and $\rho(k)$'s having a common distribution. Then the process survives if $E\frac{\delta(\lambda + \rho + \delta)}{\lambda\rho} < 1,$ while the right edge $r_t$ of the process with initial configuration $\cdots$ 111000 $\cdots$ satisfies $\lim\sup_{t\rightarrow\infty}r_t = +\infty$ if $E\log\frac{\delta(\lambda + \rho + \delta)}{\lambda\rho} < 0.$ If the environment is deterministic and periodic with period $p$, we prove survival if $\Pi^p_{k=1}\frac{\delta(k)\lbrack\lambda(k) + \rho(k - 1) + \delta(k)\rbrack}{\lambda(k)\rho(k - 1)} < 1$ and $\Pi^p_{k=1}\frac{\delta(k - 1)\lbrack\lambda(k) + \rho(k - 1) + \delta(k - 1)\rbrack}{\lambda(k)\rho(k - 1)} < 1.$

#### Article information

**Source**

Ann. Probab., Volume 20, Number 2 (1992), 696-723.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176989801

**Digital Object Identifier**

doi:10.1214/aop/1176989801

**Mathematical Reviews number (MathSciNet)**

MR1159569

**Zentralblatt MATH identifier**

0754.60126

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

**Keywords**

Contact process processes in random environments

#### Citation

Liggett, Thomas M. The Survival of One-Dimensional Contact Processes in Random Environments. Ann. Probab. 20 (1992), no. 2, 696--723. doi:10.1214/aop/1176989801. https://projecteuclid.org/euclid.aop/1176989801