Asymptotic shape of the region visited by an Eulerian Walker

Details Asymptotic shape of the region visited by an Eulerian Walker Asymptotic shape of the region visited by an Eulerian Walker

2009-06-30

arxiv.org/abs/0906.5506v2

Physics

Condensed Matter

Statistical Mechanics

7 pages, 11 figures, minor revisions

Scientific paper

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, $ \sim N^{2\nu}$, shows a crossover from the Eulerian ($\nu = 1/3$) to a simple random walk ($\nu=1/2$) behaviour.

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