Asymptotic shape of the region visited by an eulerian walker

Abstract

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 N^1/3, for large N, and the width of the boundary region grows as N^α/3, with α=0.40±0.06. If we introduce stochasticity in the evolution rules, the mean-square displacement of the walker, <R_N²> ~ N^2ν, shows a crossover from the Eulerian (ν=1/3) to a simple random-walk (ν=½) behavior.