Electronic Communications in Probability

Scaling Limit of the Prudent Walk

Vincent Beffara, Sacha Friedli, and Yvan Velenik

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60G50: Sums of independent random variables; random walks 60G52: Stable processes

prudent self-avoiding walk brownian motion scaling limit ballistic behaviour ageing

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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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