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April 1999 The Second Lowest Extremal Invariant Measure of the Contact Process II
Marcia Salzano, Roberto H. Schonmann
Ann. Probab. 27(2): 845-875 (April 1999). DOI: 10.1214/aop/1022677388


We continue the investigation of the behavior of the contact process on infinite connected graphs of bounded degree. Some questions left open by Salzano and Schonmann (1997) concerning the notions of complete convergence, partial convergence and the criterion $r = s$ are answered.

The continuity properties of the survival probability and the recurrence probability are studied. These order parameters are found to have a richer behavior than expected, with the possibility of the survival probability being discontinuous at or above the threshold for survival. A condition which guarantees the continuity of the survival probability above the survival point is introduced and exploited. The recurrence probability is shown to always be left-continuous above the recurrence point, and a necessary and sufficient condition for its right-continuity is introduced and exploited. It is shown that for homogeneous graphs the survival probability can only be discontinuous at the survival point, and the recurrence probability can only be discontinuous at the recurrence point.

For graphs which are obtained by joining a finite number of severed homogeneous trees by means of a finite number of vertices and edges, the survival point, the recurrence point and the discontinuity points of the survival and recurrence probabilities are located.


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Marcia Salzano. Roberto H. Schonmann. "The Second Lowest Extremal Invariant Measure of the Contact Process II." Ann. Probab. 27 (2) 845 - 875, April 1999.


Published: April 1999
First available in Project Euclid: 29 May 2002

zbMATH: 0946.60088
MathSciNet: MR1698975
Digital Object Identifier: 10.1214/aop/1022677388

Primary: 60K35

Keywords: complete convergence , contact process , continuity properties , criterion r = s , critical points , ergodic behavior , Graphs , Invariant measures , order parameters , partial convergence

Rights: Copyright © 1999 Institute of Mathematical Statistics

Vol.27 • No. 2 • April 1999
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