Electronic Communications in Probability

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.

Article information

Source
Electron. Commun. Probab., Volume 15 (2010), paper no. 5, 44-58.

Dates
Accepted: 24 February 2010
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465243948

Digital Object Identifier
doi:10.1214/ECP.v15-1527

Mathematical Reviews number (MathSciNet)
MR2595682

Zentralblatt MATH identifier
1201.60029

Rights

Citation

Beffara, Vincent; Friedli, Sacha; Velenik, Yvan. Scaling Limit of the Prudent Walk. Electron. Commun. Probab. 15 (2010), paper no. 5, 44--58. doi:10.1214/ECP.v15-1527. https://projecteuclid.org/euclid.ecp/1465243948

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