A shape theorem for the orthant model

We study a particular model of random medium, called the orthant model, in general dimensions d≥2. Each site x∈Zd independently has arrows pointing to its positive neighbours x+ei, i=1,…,d with probability p and, otherwise, negative x−ei, (with 1−p). prove shape theorem for set sites reachable by following arrows, starting from origin, when is large. The argument uses subadditivity, as would be expected theorems arising first passage percolation. main difficulty overcome that primary objects are not stationary which key requirement subadditive ergodic theorem.