Asymptotic shape of the region visited by an Eulerian Walker

Abstract

We study an Eulerian walker on a square lattice, starting from an initially randomly oriented background using Monte Carlo simulations. We present evidence that, that, for 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 $N^{1/3}$, for large $N$, and the width of the boundary region grows as $N^{\alpha /3}$, with $\alpha =0.40\pm .05$. If we introduce stochasticity in the evolution rules, the mean square displacement of the walker, $<R_N^2>\sim N^{2\nu }$, shows a crossover from the Eulerian ($\nu =1/3$) to a simple random walk ($\nu =1/2$) behaviour.

7 pages, 11 figures, minor revisions

References in corpus (5)

• Pattern formation in growing sandpiles
• Limiting shapes for deterministic centrally seeded growth models
• Sticky grains do not change the universality class of isotropic sandpiles
• The Rotor-Router Model
• Self-repelling random walk with directed edges on Z

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