Asymptotic shape of the region visited by an Eulerian walker

Abstract

We study an Eulerian walker on a square lattice, starting from an initial randomly oriented background using Monte Carlo simulations. We present evidence that, for a large number of steps N , the asymptotic shape of the set of sites visited by the walker is a perfect circle. The radius of the circle increases as N1/3, for large N , and the width of the boundary region grows as Nalpha/3, with alpha=0.40+/-0.06 . If we introduce stochasticity in the evolution rules, the mean-square displacement of the walker, <RN2> approximately <RN2> approximately N2nu, shows a crossover from the Eulerian (nu=1/3) to a simple random-walk (nu=1/2) behavior.

Rotor-router walk on a semi-infinite cylinder

“…The Archimedean property is consistent with the perfect circular asymptotic shape of the cluster of visited sites. Also, it agrees with a scaling law for the number of visits to a site separated from the origin by distance x for N steps of the walker [7].…”

“…A standard approach to investigation of fluctuations of a growing surface implies a formulation in terms of the KPZ theory [10]. Kapri and Dhar [7] calculated the width W of the surface region of the rotor-router cluster on the square lattice to determine the saturation-width exponent α and the dynamical exponent z in the KPZ scaling law W ∼ L α f KP Z (t/L z ), (1.1) where L is a characteristic length of the surface, and f KP Z (x) is the scaling function which behaves as x α/z for 0 < x ≪ 1 and tends to 1 for x ≫ 1 [10]. Since for the growing two-dimensional cluster of radius R, both time t and length L are proportional to R, the expected KPZ values α = 1/2 and z = 3/2 lead to asymptotic law W ∼ R γ = R 1/3 .…”

Range and Speed of Rotor Walks on Trees

“…On combs, it is proven that the size of the range | | is of order 2/3 , and its asymptotic shape is a diamond. It is conjectured in [KD09], that on ℤ 2 , the range of uniform rotor walks is asymptotically a disk, and its size is of order 2/3 . In the recent paper [Cha18], for special cases of initial configuration of rotors on transient and vertex-transitive graphs, it is shown that the occupation rate of the rotor walk is close to the Green function of the random walk.…”

Spiral structures in the rotor-router walk

“…Monte Carlo simulations in [14] showed that the average number of visits of a site inside the disk is a linear decreasing function of its distance from the origin. The authors of [14] give the following explanation of this characteristic behavior. After a moment when two sites at different distances from the origin are visited by the rotor walk, both sites are visited equally often because of the local Euler-like organization of arrows.…”

“…The existence of the limit for the averaged ratio of radius r to angle θ which is a constant b for the purely Archimedean case, is not proven yet. A prospective value of b can be obtained from the scaling law for the average number of visits conjectured by Kapri and Dhar [14].…”