Scaling Limit of the Prudent Walk

Details Scaling Limit of the Prudent Walk Scaling Limit of the Prudent Walk

2009-02-25

arxiv.org/abs/0902.4312v4

Electron. Commun. Probab., Vol. 15 (2010) , p. 44--58

Mathematics

Probability

Better exposition, stronger claim, simpler description of the limiting process; final version, to appear in Electr. Commun. Pr

Scientific paper

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.

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