TY  - JOUR
A1  - Friedrich, Tobias
A1  - Katzmann, Maximilian
A1  - Krohmer, Anton
T1  - Unbounded Discrepancy of Deterministic Random Walks on Grids
JF  - SIAM journal on discrete mathematics
N2  - Random walks are frequently used in randomized algorithms. We study a derandomized variant of a random walk on graphs called the 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 in a remarkably similar way: 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 [Combin. Probab. Comput., 15 (2006), pp. 815-822] showed that on 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 of these results, however, assume that initially all tokens are only placed on one partition of the bipartite graph 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 >= 1 and arbitrary discrepancies l >= 0, we construct configurations that reach a discrepancy of at least l.
KW  - deterministic random walk
KW  - rotor-router model
KW  - single vertex discrepancy
Y1  - 2018
U6  - https://doi.org/10.1137/17M1131088
SN  - 0895-4801
SN  - 1095-7146
VL  - 32
IS  - 4
SP  - 2441
EP  - 2452
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  -