Sharpness versus robustness of the percolation transition in 2D contact processes

TL;DR

This paper investigates the sharpness of the percolation transition in 2D contact processes with multiple states and density-dependent infection rates, challenging previous notions of robust critical behavior.

Contribution

It proves the sharpness of the percolation transition in these models, providing new insights into their critical phenomena and contradicting earlier claims of robust criticality.

Findings

Percolation transition is sharp in the studied models.

Contradicts the idea of robust critical behavior with power law cluster sizes.

Provides rigorous proof under certain assumptions.

Abstract

We study versions of the contact process with three states, and with infections occurring at a rate depending on the overall infection density. Motivated by a model described in [17] for vegetation patterns in arid landscapes, we focus on percolation under invariant measures of such processes. We prove that the percolation transition is sharp (for one of our models this requires a reasonable assumption). This is shown to contradict a form of 'robust critical behaviour' with power law cluster size distribution for a range of parameter values, as suggested in [17].