Unbounded Discrepancy of Deterministic Random Walks on Grids

Abstract

Random walks are frequently used in randomized algorithms. We study a derandomized variant of a random walk on graphs, called rotor-router model. In this model, instead of distributing tokens randomly, each vertex serves its neighbors in a fixed deterministic order. For most setups, both processes behave remarkably similar: Starting with the same initial configuration, the number of tokens in the rotor-router model deviates only slightly from the expected number of tokens on the corresponding vertex in the random walk model. The maximal difference over all vertices and all times is called single vertex discrepancy. Cooper and Spencer (2006) showed that on $\mathbb {Z}^{d}$ the single vertex discrepancy is only a constant $c_d$. Other authors also determined the precise value of $c_d$ for $d=1,2$. All these results, however, assume that initially all tokens are only placed on one partition of the bipartite graph $\mathbb {Z}^{d}$. We show that this assumption is crucial by proving that otherwise the single vertex discrepancy can become arbitrarily large. For all dimensions $d\ge 1$ and arbitrary discrepancies $\ell \ge 0$, we construct configurations that reach a discrepancy of at least $\ell$.