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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  • O. Angel, I. Benjamini, and B. Virág. Random walks that avoid their past convex hull. Electron. Comm. Probab., 8:6–16 (electronic), 2003.
  • M. Bousquet-Mélou. Families of prudent self-avoiding walks. Preprint arXiv: 0804.4843.
  • J.C. Dethridge, T.M. Garoni, A.J. Guttmann, and I.Jensen. Prudent walks and polygons. Preprint arXiv: 0810.3137.
  • J.C. Dethridge and A.J. Guttmann. Prudent self-avoiding walks. Entropy, 10:309–318, 2008.
  • E. Duchi. On some classes of prudent walks. FPSAC '05, Taormina, Italy, 2005.
  • A.J. Guttmann. Some solvable, and as yet unsolvable, polygon and walk models. International Workshop on Statistical Mechanics and Combinatorics: Counting Complexity, 42:98–110, 2006.
  • J. Komlós, P. Major, and G. Tusnády. An approximation of partial sums of independent RV's, and the sample DF II. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 34(1):33–58, 1976.
  • G.F. Lawler and V. Limic. Random Walk: a Modern Introduction. Preliminary draft (version: February 2010).
  • S.B. Santra, W.A. Seitz, and D.J. Klein. Directed self-avoiding walks in random media. Phys. Rev. E, 63(6):067101, Part 2 JUN 2001.
  • U. Schwerdtfeger. Exact solution of two classes of prudent polygons. Preprint arXiv: 0809.5232.
  • L. Turban and J.-M. Debierre. Self-directed walk: a Monte Carlo study in three dimensions. J. Phys. A, 20:3415–3418, 1987.
  • L. Turban and J.-M. Debierre. Self-directed walk: a Monte Carlo study in two dimensions. J. Phys. A, 20:679–686, 1987.
  • M.P.W. Zerner. On the speed of a planar random walk avoiding its past convex hull. Ann. Inst. H. Poincaré Probab. Statist., 41(5):887–900, 2005.