| The d-dimensional Hamming torus is the graph whose vertices are all of the integer points inside an a1 n x a2n x ··· x adn box in Rd (for constants a1, ..., ad > 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 = ln , then there exists lambdac > 0, which is the positive root of a degree d polynomial whose coefficients depend on a1, ..., ad, such that for lambda < lambdac the largest component has O(log n) vertices (a.a.s. as n → infinity), and for lambda > lambda c the largest component has (1 - q)lambda (producti ai) nd -1 + o(nd -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 = clognn , then when c < d-1a i the site subgraph of the Hamming torus is not connected, and when c > d-1a i the subgraph is connected (a.a.s.). We also show that the subgraph is connected precisely when it contains no isolated vertices. Finally, we present computer simulations of the model, and discuss the strengths and shortcomings of our theorems. |
