Scaling Limit of the Prudent Walk

Vincent Beffara (UMPA)
Sacha Friedli (UFMG)
Yvan Velenik (Université de Genève)

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.

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Pages: 44-58

Publication Date: February 24, 2010

DOI: 10.1214/ECP.v15-1527

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