Scaling Limit of the Prudent Walk

TL;DR

This paper establishes the scaling limit of the prudent walk on a square lattice, showing it converges to a process involving Brownian motion and determining its asymptotic speed as 3/7.

Contribution

It provides the first rigorous description of the scaling limit for the prudent walk, linking it to Brownian motion and explicitly calculating its asymptotic speed.

Findings

Scaling limit described by a process involving Brownian motion.

Asymptotic speed of the walk is 3/7 in L^1-norm.

Convergence of the prudent walk to the specified process.

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) = s_1 theta^+(3u/7) e_1 + s_2 theta^-(3u/7) e_2, where e_1, e_2 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 s_1, s_2 are two random signs. In particular, the asymptotic speed of the walk is well-defined in the L^1-norm and equals 3/7.