The Annals of Probability

The second lowest extremal invariant measure of the contact process

Marcia Salzano and Roberto H. Schonmann

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We study the ergodic behavior of the contact process on infinite connected graphs of bounded degree. We show that the fundamental notion of complete convergence is not as well behaved as it was thought to be. In particular there are graphs for which complete convergence holds in any number of separated intervals of values of the infection parameter and fails for the other values of this parameter. We then introduce a basic invariant probability measure related to the recurrence properties of the process, and an associated notion of convergence that we call “partial convergence.” This notion is shown to be better behaved than complete convergence, and to hold in certain cases in which complete convergence fails. Relations between partial and complete convergence are presented, as well as tools to verify when these properties hold. For homogeneous graphs we show that whenever recurrence takes place (i.e., whenever local survival occurs) there are exactly two extremal invariant measures.

Article information

Ann. Probab., Volume 25, Number 4 (1997), 1846-1871.

First available in Project Euclid: 7 June 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Contact process graphs invariant measures ergodic behavior critical points complete convergence partial convergence


Salzano, Marcia; Schonmann, Roberto H. The second lowest extremal invariant measure of the contact process. Ann. Probab. 25 (1997), no. 4, 1846--1871. doi:10.1214/aop/1023481114.

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