In this paper we show how to construct an algorithm to sample the stationary distribution of a random walk over a grid with forbidden arcs. This algorithm combines the rejection method and coupling from the past of a set of trajectories of the Markov chain that generalizes the classical sandwich approach. We also provide a complexity analysis of this approach in several cases showing a coupling time that is logarithmic in the size of the grid, when no arc is forbidden, and an experimental study of its performance.