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 Z(2) x K-n(2), where K-n 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 Z(2) x K-n. The community structure of the product graphs connects our process to a heterogeneous bootstrap percolation on Z(2). This natural relation with a generalization of polluted bootstrap percolation is the leading theme in our analysis.