Asymptotic Analysis of a Drop-Push Model For Percolation

In this article, we study a type of a one dimensional percolation model whose basic features include a sequential dropping of particles on a substrate followed by their transport via a pushing mechanism (see [S. N. Majumdar and D. S. Dean, Phys. Rev. Ltt. A 11, 89 (2002)]). Consider an empty one dimensional lattice with n empty sites and periodic boundary conditions (as a necklace with n rings). Imagine then the particles which drop sequentially on this lattice, uniformly at random on one of the n sites. Letting a site can settles at most one particle, if a particle drops on an empty site, it stick there and otherwise the particle moves according to a symmetric random walk until it takes place in the first empty site it meet. We study here, the asymptotic behavior of the arrangement of empty sites and of the total displacement of all particles as well as the partial displacement of some particles.