Sharpness for monotone absorbing Interacting Particle Systems

We prove a sharpness result for the dynamics of finite-range Interacting Particle Systems (IPS) on $\{0,1{\}}^{{\mathbb{Z}}^d}$, which generalizes to a whole class of IPS the sharpness result for the phase transition of the contact process obtained by Bezuidenhout and Grimmett~\cite{BezuidenhoutGrimmett1991}. More precisely, starting from an IPS that is monotone, ergodic, and which admits the all-zero configuration as an absorbing state, we prove that there exists an arbitrarily small perturbation of the dynamics which leads to an \emph{exponentially} ergodic IPS. This also extends the sharpness result previously established for (discrete-time) probabilistic cellular automata in \cite{Har} to the continuous-time setting of IPS.