Asymptotic shape of the region visited by an Eulerian Walker

TL;DR

This study investigates the asymptotic shape of the region visited by an Eulerian walker on a lattice, revealing a circular shape with radius scaling as N^{1/3} and boundary width growth, with stochasticity inducing a crossover to random walk behavior.

Contribution

The paper provides the first detailed analysis of the asymptotic shape and scaling behavior of Eulerian walkers, including effects of stochasticity on their displacement.

Findings

Visited region approaches a perfect circle for large N.

Radius of the visited region scales as N^{1/3}.

Stochasticity causes a crossover from Eulerian to random walk behavior.

Abstract

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.