## The Annals of Probability

- Ann. Probab.
- Volume 2, Number 6 (1974), 969-988.

### Contact Interactions on a Lattice

#### Abstract

Let $\{\xi_t\}$ be a Markov process whose values are subsets of $Z_d$, the $d$-dimensional integers. Put $\xi_t(x) = 1$ if $x \in \xi_t$ and 0 otherwise. The transition intensity for a change in $\xi_t(x)$ depends on $\{\xi_t(y), y$ a neighbor of $x\}$. The chief concern is with "contact processes," where $\xi_t(x)$ can change from 0 to 1 only if $\xi_t(y) = 1$ for some $y$ neighboring $x$. Let $p_t(\xi) = \operatorname{Prob} \{\xi_t \neq \varnothing \mid \xi_0 = \xi\}$. Under appropriate conditions, $p_t$ is increasing, subadditive, or submodular in $\xi$. In the case of contact processes, conditions are giving implying that $p_\infty(\xi) = 0$ for all finite $\xi$, or that the contrary is true. In other cases conditions for ergodicity are given.

#### Article information

**Source**

Ann. Probab. Volume 2, Number 6 (1974), 969-988.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996493

**Digital Object Identifier**

doi:10.1214/aop/1176996493

**Mathematical Reviews number (MathSciNet)**

MR356292

**Zentralblatt MATH identifier**

0334.60052

**JSTOR**

links.jstor.org

**Subjects**

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

**Keywords**

Contact interaction birth-death interaction subadditivity ergodicity

#### Citation

Harris, T. E. Contact Interactions on a Lattice. Ann. Probab. 2 (1974), no. 6, 969--988. doi:10.1214/aop/1176996493. https://projecteuclid.org/euclid.aop/1176996493.