  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.
