In this paper we establish a decoupling feature of the random interlacement process ${ℐ}^u\subset {\mathbb{Z}}^d$ at level $u$, $d\ge 3$. Roughly speaking, we show that observations of ${ℐ}^u$ restricted to two disjoint subsets $A_1$ and $A_2$ of ${\mathbb{Z}}^d$ are approximately independent, once we add a sprinkling to the process ${ℐ}^u$ by slightly increasing the parameter $u$. Our results differ from previous ones in that we allow the mutual distance between the sets $A_1$ and $A_2$ to be much smaller than their diameters. We then provide an important application of this...

We consider a broad class of stochastic lattice predator-prey models whose main features are overviewed. In particular, this article aims at drawing a picture of the influence of spatial fluctuations, which are not accounted for by the deterministic rate equations, on the properties of the stochastic models. Here, we outline the robust scenario obeyed by most of the lattice predator-prey models with an interaction à la Lotka-Volterra. We also show how a drastically different behavior can emerge...

In this article, we consider the following model of self-avoiding walk: the probability of a self-avoiding trajectory $\gamma$ between two points on the boundary of a finite subdomain of ${\mathbb{Z}}^d$ is proportional to ${\mu }^{-\text{length}\left(\gamma \right)}$. When $\mu$ is supercritical (i.e. $\mu lt;{\mu }_c$ where ${\mu }_c$ is the connective constant of the lattice), we show that the random trajectory becomes space-filling when taking the scaling limit.