Sharpness for monotone absorbing Interacting Particle Systems
Jean Bérard, Barbara Dembin, Laure Marêché
TL;DR
The paper proves a sharpness result for monotone absorbing interacting particle systems on $\{0,1\}^{\mathbb Z^d}$, showing that any ergodic IPS in this class can be perturbed by an arbitrarily small amount to become exponentially ergodic. The authors develop a continuous-time OSSS framework for Poisson functionals, grounded in a graphical construction and pivotality analysis, to relate perturbations to exponential decay of correlations via a Russo-type derivative of $\theta_T(h)$. They introduce a backward-in-time influence set $I_t^T$ and establish localization bounds, then formulate and apply a continuous OSSS inequality to derive a differential inequality that forces exponential decay for any positive perturbation parameter $h$. Consequently, the perturbed dynamics $L^\varepsilon$ inherit exponential ergodicity, confirming a sharpness phenomenon that extends Bezuidenhout-Grimmett style results from the contact process to a broad class of continuous-time IPS. The work also clarifies the topological structure of the set of ergodic generators within this class and connects IPS sharpness to percolation-style techniques via OSSS.
Abstract
We prove a sharpness result for the dynamics of finite-range Interacting Particle Systems (IPS) on $\{0,1\}^{\Z^d}$, 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.