Open Access
October 1997 The second lowest extremal invariant measure of the contact process
Marcia Salzano, Roberto H. Schonmann
Ann. Probab. 25(4): 1846-1871 (October 1997). DOI: 10.1214/aop/1023481114


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.


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Marcia Salzano. Roberto H. Schonmann. "The second lowest extremal invariant measure of the contact process." Ann. Probab. 25 (4) 1846 - 1871, October 1997.


Published: October 1997
First available in Project Euclid: 7 June 2002

zbMATH: 0903.60085
MathSciNet: MR1487439
Digital Object Identifier: 10.1214/aop/1023481114

Primary: 60K35

Keywords: complete convergence , contact process , critical points , ergodic behavior , Graphs , Invariant measures , partial convergence

Rights: Copyright © 1997 Institute of Mathematical Statistics

Vol.25 • No. 4 • October 1997
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