Proportionate growth in patterns formed in the rotor-router model

Abstract

We study the growing patterns formed in the rotor-router model by adding N walkers at the centre of an L × L two-dimensional square lattice, starting with a periodic background of arrows, and relaxing to a stable configuration. The pattern is made of a large number of triangular and quadrilateral regions, where in each region all arrows point in the same direction. We show that the pattern formed by arrows which have been rotated at least one full circle may be described in terms of a tiling of the plane by squares of different sizes. The sizes of these squares, and the size of the pattern, grow linearly with N for 1 ≪ N < 2L. We use the Brooks–Smith–Stone–Tutte theorem relating tilings of squares by smaller squares to resistor networks, to determine the exact relative sizes of these tiles for large N. The scaling limit of the number of visits φ(ξ, η) as a function of the scaled position (ξ, η) is also determined. We also present numerical evidence that the deviations of the sizes of the different squares in the tiling from the asymptotic linear growth law are always O(1), and are quasiperiodic functions of N.