Scaling Limit of the Prudent Walk
Abstract
We describe the scaling limit of the nearest neighbour prudent walk on the square lattice, 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) = s1 theta+(3u/7) e1 + s2 theta-(3u/7) e2, where e1, e2 is the canonical basis, theta+(t), resp. theta-(t), is the time spent by a one-dimensional Brownian motion above, resp. below, 0 up to time t, and s1, s2 are two random signs. In particular, the asymptotic speed of the walk is well-defined in the L1-norm and equals 3/7.
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