One-dimensional random walks with self-blocking immigration

We consider a system of independent one-dimensional random walkers where new particles are added at the origin at fixed rate whenever there is no older particle present at the origin. A Poisson ansatz leads to a semi-linear lattice heat equation and predicts that starting from the empty configuration the total number of particles grows as c.t log t. We confirm this prediction and also describe the asymptotic macroscopic profile of the particle configuration.