Subcritical contact processes seen from a typical infected site

TL;DR

This paper investigates the long-term behavior of subcritical contact processes from a typical infected site, establishing uniqueness of eigenmeasures and analyzing the dependence of growth rates on recovery parameters.

Contribution

It proves the uniqueness of spatially homogeneous eigenmeasures for subcritical contact processes on countable groups and derives a formula for the growth rate's derivative.

Findings

Unique eigenmeasure in the subcritical regime

Continuous differentiability of the exponential growth rate

Explicit formula for the derivative of the growth rate

Abstract

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.