First we go through the differences between the classical random walk and the quantum random walk (QRW) on Z. We generalize these differences briefly and then move to define a QRW on Z^2. From there we turn to generating functions to keep track of the location probabilities. We use these generating functions to extract asymptotic information about the region on which the probability of the walk occurring is not exponentially decaying. Lastly we show some comparison pictures of actual walks with the region of non-exponential decay and see some surprising similarities.