Electronic Journal of Probability

Scaling limit of the uniform prudent walk

Nicolas Pétrélis, Rongfeng Sun, and Niccolò Torri

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1504749661

Digital Object Identifier
doi:10.1214/17-EJP87

Mathematical Reviews number (MathSciNet)
MR3698735

Zentralblatt MATH identifier
1377.82028

Subjects
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]

Keywords
self-avoiding walk prudent walk scaling limits

Rights
Creative Commons Attribution 4.0 International License.

Citation

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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References

  • [1] N.R. Beaton and G.K. Iliev, Two-sided prudent walks: a solvable non-directed model of polymer adsorption, J. Stat. Mech. Theory Exp. (2015), no. 9, P09014, 23.
  • [2] V. Beffara, S. Friedli, and Y. Velenik, Scaling limit of the prudent walk, Electronic Communications in Probability 15 (2010), 44–58.
  • [3] M. Bousquet-Mélou, Families of prudent self-avoiding walks, Journal of combinatorial theory Series A 117 (2010), no. 3, 313 – 344.
  • [4] J.C. Dethridge and A.J. Guttmann, Prudent self-avoiding walks, Entropy.
  • [5] G. Giacomin, Random polymer models, Imperial College Press, World Scientific, 2007.
  • [6] N. Pétrélis and N. Torri, Collapse transition of the interacting prudent walk, arXiv:1610.07542, 2016.
  • [7] S.B. Santra, W.A. Seitz, and D.J. Klein, Directed self-avoiding walks in random media, Phys. Rev. E 63 (2001), no. 6, 067101.
  • [8] L. Turban and J.M. Debierre, Self-directed walk: a monte carlo study in three dimensions, J. Phys. A 20 (1987), 3415 – 3418.
  • [9] L. Turban and J.M. Debierre, Self-directed walk: a monte carlo study in two dimensions, J. Phys. A 20 (1987), 679 – 686.