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 $\theta$ 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 $\mathbb{Z}^2\times K_n^2$, 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 $\theta$ exhibits a sharp phase transition, while odd $\theta$ yields a gradual percolation transition. We also establish a gradual transition for bootstrap percolation on $\mathbb{Z}^2\times K_n$. The main tool is heterogeneous bootstrap percolation on $\mathbb{Z}^2$.