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The d-dimensional Hamming torus is the graph whose vertices are all of theinteger points inside an a_1 n X a_2 n X ... X a_d n box in R^d (for constantsa_1, ..., a_d > 0), and whose edges connect all vertices within Hammingdistance one. We study the size of the largest connected component of thesubgraph generated by independently removing each vertex of the Hamming toruswith 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 whosecoefficients depend on a_1, ..., a_d, such that for \lambda < \lambda_c thelargest 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 theemergence of a giant component in the random edge subgraph of the Hammingtorus. 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 thesubgraph is connected precisely when it contains no isolated vertices.

Emergence of a Giant Component in Random Site Subgraphs of a d-Dimensional Hamming Torus