  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.
