The Minesweeper Game: Percolation and Complexity

We study a model motivated by the minesweeper game. In this model one starts with the percolation of mines on the lattice Zd, and then tries to find an infinite path of mine-free sites. At every recovery of a free site, the player is given some information on the sites adjacent to the current site. We compare the parameter values for which there exists a strategy such that the process survives to the critical parameter of ordinary percolation. We then derive improved bounds for these values for the same process, when the player has some complexity restrictions in computing his moves. Finally, we discuss some monotonicity issues which arise naturally for this model.