Asymptotic shape of the region visited by an Eulerian Walker

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.