Deterministic Random Walks on the Integers

Jim Propp's P-machine, also known as the "rotor router model" is a simple deterministic process that simulates a random walk on a graph. Instead of distributing chips to randomly chosen neighbors, it serves the neighbors in a fixed order. We investigate how well this process simulates a random walk. For the graph being the infinite path, we show that, independent of the starting configuration, at each time and on each vertex, the number of chips on this vertex deviates from the expected number of chips in the random walk model by at most a constant c_1, which is approximately 2.29. For intervals of length L, this improves to a difference of O(log L), for the L_2 average of a contiguous set of intervals even to O(sqrt{log L}). All these bounds are tight

Deterministic Random Walks on the Integers

We analyze the one-dimensional version of Jim Propp's $P$-machine, a simple deterministic process that simulates a random walk on $\mathbb{Z}$. The "output'' of the machine is astonishingly close to the expected behavior of a random walk, even on long intervals of space and time

Deterministic Random Walks on Regular Trees

Jim Propp's rotor router model is a deterministic analogue of a random walk on a graph. Instead of distributing chips randomly, each vertex serves its neighbors in a fixed order. Cooper and Spencer (Comb. Probab. Comput. (2006)) show a remarkable similarity of both models. If an (almost) arbitrary population of chips is placed on the vertices of a grid $\Z^d$ and does a simultaneous walk in the Propp model, then at all times and on each vertex, the number of chips on this vertex deviates from the expected number the random walk would have gotten there by at most a constant. This constant is independent of the starting configuration and the order in which each vertex serves its neighbors. This result raises the question if all graphs do have this property. With quite some effort, we are now able to answer this question negatively. For the graph being an infinite $k$-ary tree ($k \ge 3$), we show that for any deviation $D$ there is an initial configuration of chips such that after running the Propp model for a certain time there is a vertex with at least $D$ more chips than expected in the random walk model. However, to achieve a deviation of $D$ it is necessary that at least $\exp(\Omega(D^2))$ vertices contribute by being occupied by a number of chips not divisible by $k$ at a certain time.Comment: 15 pages, to appear in Random Structures and Algorithm

Discrete analogue computing with rotor-routers

Rotor-routing is a procedure for routing tokens through a network that can implement certain kinds of computation. These computations are inherently asynchronous (the order in which tokens are routed makes no difference) and distributed (information is spread throughout the system). It is also possible to efficiently check that a computation has been carried out correctly in less time than the computation itself required, provided one has a certificate that can itself be computed by the rotor-router network. Rotor-router networks can be viewed as both discrete analogues of continuous linear systems and deterministic analogues of stochastic processes.Comment: To appear in Chaos Special Focus Issue on Intrinsic and Designed Computatio