  We study diffusive lattice gases with local injection of particles, namely we
assume that whenever the origin becomes empty, a new particle is immediately
injected into the origin. We consider two lattice gases: a symmetric simple
exclusion process and random walkers. The interplay between the injection
events and the positions of the particles already present implies an effective
collective interaction even for the ostensibly non-interacting random walkers.
We determine the average total number of particles entering into the initially
empty system. We also compute the average total number of distinct sites
visited by all particles, and discuss the shape of the visited domain and the
statistics of visits.
