Electronic Journal of Probability

Scaling limit of the uniform prudent walk

Nicolas Pétrélis, Rongfeng Sun, and Niccolò Torri

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We study the 2-dimensional uniform prudent self-avoiding walk, which assigns equal probability to all nearest-neighbor self-avoiding paths of a fixed length that respect the prudent condition, namely, the path cannot take any step in the direction of a previously visited site. The uniform prudent walk has been investigated with combinatorial techniques in [3], while another variant, the kinetic prudent walk has been analyzed in detail in [2]. In this paper, we prove that the $2$-dimensional uniform prudent walk is ballistic and follows one of the $4$ diagonals with equal probability. We also establish a functional central limit theorem for the fluctuations of the path around the diagonal.

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 66, 19 pp.

Received: 21 February 2017
Accepted: 26 July 2017
First available in Project Euclid: 7 September 2017

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Zentralblatt MATH identifier

Primary: 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]
Secondary: 60F05: Central limit and other weak theorems 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

self-avoiding walk prudent walk scaling limits

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Pétrélis, Nicolas; Sun, Rongfeng; Torri, Niccolò. Scaling limit of the uniform prudent walk. Electron. J. Probab. 22 (2017), paper no. 66, 19 pp. doi:10.1214/17-EJP87. https://projecteuclid.org/euclid.ejp/1504749661

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