Pseudorandom walks and chip firing games

We study two deterministic analogues of random walks. The first is the chip-firing game, a single player game played by moving chips around a directed graph, popularised by Björner and Lovász. We find an efficient simulation of boolean circuits and Turing machines using instances of the chip-firing game - after assigning a fixed strategy to the player. The second is the Propp machine, or the rotor router model, a quasirandom model intro- duced by Priezzhev. We improve results of Kijima et al. and show a bound of O(m) on the discrepancy of this process from a random walk on d-regular graphs with m edges.