Goldbug variations

Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile

“…In higher dimensions, the model can be defined analogously by repeatedly cycling the rotors through an ordering of the 2d cardinal directions in Z d . Jim Propp observed from simulations in two dimensions that the regions A n are extraordinarily close to circular, and asked why this was so [7,13,19]. Despite the impressive empirical evidence for circularity, the best result known until now [14] says only that if A n is rescaled to have unit volume, the volume of the symmetric difference of A n with a ball of unit volume tends to zero as a power of n, as n ↑ ∞.…”

BML revisited: Statistical physics, computer simulation, and probability

“…In fact, it has been shown that during the evolution to the final state a rotor-router swarm can realize configurations from the abelian sandpile model [22]. (Of course, this intermediate behavior says nothing about the final structure.)…”

“…We could have introduced these particles one-at-a-time, requiring that the predecessor particle complete its walk before the next particle is injected. Alternately, we could have injected all N particles at the same time and allowed them to walk in parallel (called a "rotor-router swarm" in [22]). In the latter case, whenever a site is occupied by more than one particle, one of those particles is chosen uniformly at random and updated.…”

“…22. It is essentially a deterministic version of Internal Diffusion Limited Aggregation (IDLA) [23,24].…”

Deterministic walks with choice

“…Rotor-router walks have been of particular interest recently. Therein, the walker is routed through nodes via some cyclic or other order (see for instance [2,3,4,5,7,8,11,14]). Here, we consider deterministic choice based not only on recent exits, but recent entries, as well.…”