Bootstrap percolation on the Hamming torus

The Hamming torus of dimension d is the graph with vertices {1,.,n}d and an edge between any two vertices that differ in a single coordinate. Bootstrap percolation with threshold . starts with a random set of open vertices, to which every vertex belongs independently with probability p, and at each time step the open set grows by adjoining every vertex with at least . open neighbors. We assume that n is large and that p scales as n.. for some .>1, and study the probability that an i-dimensional subgraph ever becomes open. For large ., we prove that the critical exponent . is about 1+d/. for i=1, and about 1+2/.+.(..3/2) for i.2. Our small . results are mostly limited to d=3, where we identify the critical . in many cases and, when .=3, compute exactly the critical probability that the entire graph is eventually open.