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

Abstract

Bootstrap percolation on a graph is a deterministic process that iteratively enlarges a set of occupied sites by adjoining points with at least θ 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 × K 2 n , 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 θ exhibits a sharp phase transition, while odd θ yields a gradual percolation transition. We also establish a gradual transition for bootstrap percolation on Z 2 × K n . The main tool is heterogeneous bootstrap percolation on Z 2 . IntroductionSpread of signals -information, say, or infection -on graphs with community structure has attracted interest in the mathematical literature recently [Schi, BL, Lal, LZ, Siv]. The idea is that any single community is densely connected, while the connections between communities are much more sparse. This naturally leads to multiscale phenomena, as the spread of the signal within a community is much faster then between different communities. Often, communities are modeled as cliques, i.e., the intra-community graph is complete, but in other cases some close-knit structure is assumed. By contrast, the inter-community graph may, for example, impose spatial proximity as a precondition for connectivity. See [Sil + ] for an applications-oriented recent survey.The principal graph under study in this paper is G = Z 2 × K 2 n , the Cartesian product between the lattice Z 2 and two copies of the complete graph K n on n points. Thus "community" consists of "individuals" determined by two characteristics, and two individuals within the community only communicate if they have one of the characteristics in common. Between the communities, communication is between like individuals that are also neighbors in the lattice. For comparison, we also address the case where each community is a clique, that is, the graph Z 2 × K n .