Bootstrap percolation on the product of the two-dimensional lattice with a Hamming square

Bootstrap percolation on a graph is a deterministic process that iteratively enlarges a set of occupied sites by adjoining points with at least . occupied neighbors. The initially occupied set is random, given by a uniform product measure with a low density p. Our main focus is on this process on the product graph .2.K2n, where Kn is a complete graph. We investigate how p scales with n so that a typical site is eventually occupied. Under critical scaling, the dynamics with even . exhibits a sharp phase transition, while odd . yields a gradual percolation transition. We also establish a gradual transition for bootstrap percolation on .2.Kn. The community structure of the product graphs connects our process to a heterogeneous bootstrap percolation on. This natural relation with a generalization of polluted bootstrap percolation is the leading theme in our analysis.