Statistics of largest cluster growth through constant rate random filling of lattices

In this paper we consider a percolation model where the probability p for a site to be occupied increases linearly in time, from 0 to 1. We analyze the way the largest cluster grows in time, and in particular, we study the statistics of the "jumps" in the mass of the largest cluster, and of the time delay between those events. Different critical behaviors are observed below and above the percolation threshold. We propose a theoretical analysis, and we check our results against direct numerical simulations.