The d-dimensional Hamming torus is the graph whose vertices are all of the integer points inside an a_1 n X a_2 n X ... X a_d n box in R^d (for constants a_1, ..., a_d > 0), and whose edges connect all vertices within Hamming distance one. We study the size of the largest connected component of the subgraph generated by independently removing each vertex of the Hamming torus with probability 1-p. We show that if p=\lambda / n, then there exists \lambda_c > 0, which is the positive root of a degree d polynomial whose coefficients depend on a_1, ..., a_d, such that for \lambda < \lambda_c the largest component has O(log n) vertices (a.a.s. as n \to \infty), and for \lambda > \lambda_c the largest component has (1-q) \lambda (\prod_i a_i) n^{d-1} + o(n^{d-1}) vertices and the second largest component has O(log n) vertices (a.a.s.). An implicit formula for q < 1 is also given. Surprisingly, the value of \lambda_c that we find is distinct from the critical value for the emergence of a giant component in the random edge subgraph of the Hamming torus. Additionally, we show that if p = c log n / n, then when c < (d-1) / (\sum a_i) the site subgraph of the Hamming torus is not connected, and when c > (d-1) / (\sum a_i) the subgraph is connected (a.a.s.). We also show that the subgraph is connected precisely when it contains no isolated vertices.

We initiate the study of general neighborhood growth dynamics on two dimensional Hamming graphs. The decision to add a point is made by counting the currently occupied points on the horizontal and the vertical line through it, and checking whether the pair of counts lies outside a fixed Young diagram. We focus on two related extremal quantities. The first is the size of the smallest set that eventually occupies the entire plane. The second is the minimum of an energy-entropy functional that comes from the scaling of the probability of eventual full occupation versus the density of the initial product measure within a rectangle. We demonstrate the existence of this scaling and study these quantities for large Young diagrams.

The Hamming torus of dimension $d$ is the graph with vertices $\{1,\dots,n\}^d$ and an edge between any two vertices that differ in a single coordinate. Bootstrap percolation with threshold $\theta$ 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 $\theta$ open neighbors. We assume that $n$ is large and that $p$ scales as $n^{-\alpha}$ for some $\alpha>1$, and study the probability that an $i$-dimensional subgraph ever becomes open. For large $\theta$, we prove that the critical exponent $\alpha$ is about $1+d/\theta$ for $i=1$, and about $1+2/\theta+\Theta(\theta^{-3/2})$ for $i\ge2$. Our small $\theta$ results are mostly limited to $d=3$, where we identify the critical $\alpha$ in many cases and, when $\theta=3$, compute exactly the critical probability that the entire graph is eventually open.