# Asymptotic shape of the region visited by an Eulerian Walker - Condensed Matter > Statistical Mechanics

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Abstract: We study an Eulerian walker on a square lattice, starting from an initiallyrandomly oriented background using Monte Carlo simulations. We present evidencethat, that, for large number of steps $N$, the asymptotic shape of the set ofsites visited by the walker is a perfect circle. The radius of the circleincreases as $N^{1-3}$, for large $N$, and the width of the boundary regiongrows as $N^{\alpha - 3}$, with $\alpha = 0.40 \pm .05$. If we introducestochasticity in the evolution rules, the mean square displacement of thewalker, $\sim N^{2 u}$, shows a crossover from the Eulerian $u= 1-3$ to a simple random walk $u=1-2$ behaviour.

Autor: Rajeev Kapri, Deepak Dhar

Fuente: https://arxiv.org/