Decaying correlations for the supercritical contact process conditioned on survival

Marta Fiocco, Willem R. Van Zwet
2003 Bernoulli  
A d-dimensional contact process is a simplified model for the spread of a biological organism or an infection on the lattice Zd. At each time t 2:: 0, every point of the lattice (or site) is either infected or healthy. As time passes, a healthy site is infected with Poisson rate A by each of its 2d immediate neighbors which is itself infected; an infected site recovers and becomes healthy with Poisson rate 1. The processes involved are independent. If the process starts with a set A C Zd of
more » ... a set A C Zd of infected sites at time t = 0, then the infection continues forever with a positive probability iff A exceeds a certain critical value. Such a process is called supercritical. Consider the supercritical contact process starting with a single infected site at the origin, conditioned on surviving forever. We develop a technique for embedding this conditional process for large t in a contact process starting at a large time s with all sites of the lattice infected. This allows us to show that the covariances for the conditional process fall off faster than any negative power of the distance, provided that this distance is at most of the order t. The results obtained in this paper will enable us to study the statistical problem of estimating the parameter A. This will be the subject of a companion paper Fiocco and van Zwet (1999) .
doi:10.3150/bj/1066418877 fatcat:zhab34kiivaxvnimbwotbnbgxy