The truncation property and continuity for the long-range contact process on $\mathbb{Z}^d$

TL;DR

This paper extends classical results on contact processes to long-range interactions on $ abla^d$, establishing conditions for supercriticality preservation after truncation and proving continuity of the non-recovery probability.

Contribution

It adapts renormalization techniques to long-range contact processes, providing new conditions for supercriticality and continuity results.

Findings

Supercriticality is preserved after truncation under certain decay conditions.

The probability of never recovering is continuous for parameters satisfying the truncation property.

Generalizes finite-range results to long-range interaction settings.

Abstract

We consider a general class of contact processes on $\mathbb{Z}^d$ with potentially long-range interactions. By adapting well established renormalization arguments to the long-range setting we extend by now classical results for finite-range processes to this more general setting. Particularly, we provide general conditions on the decay of the interactions under which a supercritical process remains supercritical after truncation of the interaction parameter at a sufficiently large distance. Further, for the family of parameters satisfying this latter truncation property, we conclude that the probability of the process to never recover is continuous.