Subcritical contact processes seen from a typical infected site

Details Subcritical contact processes seen from a typical infected site Subcritical contact processes seen from a typical infected site

2011-10-21

arxiv.org/abs/1110.4777v1

Mathematics

Probability

35 pages

Scientific paper

What is the long-time behavior of the law of a contact process started with a single infected site, distributed according to counting measure on the lattice? This question is related to the configuration as seen from a typical infected site and gives rise to the definition of so-called eigenmeasures, which are possibly infinite measures on the set of nonempty configurations that are preserved under the dynamics up to a multiplicative constant. In this paper, we study eigenmeasures of contact processes on general countable groups in the subcritical regime. We prove that in this regime, the process has a unique spatially homogeneous eigenmeasure. As an application, we show that the exponential growth rate is continuously differentiable and strictly decreasing as a function of the recovery rate, and we give a formula for the derivative in terms of the eigenmeasures of the contact process and its dual.

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