Survival of contact processes on the hierarchical group

TL;DR

This paper investigates contact processes on hierarchical groups, establishing conditions under which the process survives or dies out, and introduces a novel renormalization technique involving Markov functionals.

Contribution

It provides new criteria for phase transitions in contact processes on hierarchical groups, utilizing a coupling argument and a novel renormalization approach with Markov functionals.

Findings

Critical recovery rate is zero if infection rates decay too fast.

Sufficient conditions for nontrivial phase transition are derived.

A new renormalization technique using Rogers and Pitman's Markov functionals is introduced.

Abstract

We consider contact processes on the hierarchical group, where sites infect other sites at a rate depending on their hierarchical distance, and sites become healthy with a constant recovery rate. If the infection rates decay too fast as a function of the hierarchical distance, then we show that the critical recovery rate is zero. On the other hand, we derive sufficient conditions on the speed of decay of the infection rates for the process to exhibit a nontrivial phase transition between extinction and survival. For our sufficient conditions, we use a coupling argument that compares contact processes on the hierarchical group with freedom two with contact processes on a renormalized lattice. An interesting novelty in this renormalization argument is the use of a result due to Rogers and Pitman on Markov functionals.