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2010 Scaling Limit of the Prudent Walk
Vincent Beffara, Sacha Friedli, Yvan Velenik
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Electron. Commun. Probab. 15: 44-58 (2010). DOI: 10.1214/ECP.v15-1527

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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Vincent Beffara. Sacha Friedli. Yvan Velenik. "Scaling Limit of the Prudent Walk." Electron. Commun. Probab. 15 44 - 58, 2010. https://doi.org/10.1214/ECP.v15-1527

Information

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

zbMATH: 1201.60029
MathSciNet: MR2595682
Digital Object Identifier: 10.1214/ECP.v15-1527

Subjects:
Primary: 60F17
Secondary: 60G50 , 60G52

Keywords: Ageing , ballistic behaviour , Brownian motion , prudent self-avoiding walk , Scaling limit

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