Scaling Limit of the Prudent Walk

Abstract

We describe the scaling limit of the nearest neighbour prudent walk on $Z^2$, which performs steps uniformly in directions in which it does not see sites already visited. We show that the scaling limit is given by the process $Z_u = \int_0^{3u/7} ( \sigma_1 1_{W(s)\geq 0}\vec{e}_1 + \sigma_2 1_{W(s)\geq 0}\vec{e}_2 ) ds$, $u \in [0,1]$, where $W$ is the one-dimensional Brownian motion and $\sigma_1, \sigma_2$ two random signs. In particular, the asymptotic speed of the walk is well-defined in the $L^1$-norm and equals 3/7.