Asymptotic shape of the region visited by an Eulerian walker.

We study an Eulerian walker on a square lattice, starting from an initial randomly oriented background using Monte Carlo simulations. We present evidence that, for a 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 N1/3, for large N , and the width of the boundary region grows as Nalpha/3, with alpha=0.40+/-0.06 . If we introduce stochasticity in the evolution rules, the mean-square displacement of the walker, approximately approximately N2nu, shows a crossover from the Eulerian (nu=1/3) to a simple random-walk (nu=1/2) behavior.