Electronic Journal of Probability

Multi-level pinning problems for random walks and self-avoiding lattice paths

Pietro Caputo, Fabio Martinelli, and Fabio Toninelli

Full-text: Open access


We consider a generalization of the classical pinning problem for integer-valued random walks conditioned to stay non-negative. More specifically, we take pinning potentials of the form $\sum_{j\geq 0}\epsilon_j N_j$, where $N_j$ is the number of visits to the state $j$ and $\{\epsilon_j\}$ is a non-negative sequence. Partly motivated by similar problems for low-temperature contour models in statistical physics, we aim at finding a sharp characterization of the threshold of the wetting transition, especially in the regime where the variance $\sigma^2$ of the single step of the random walk is small. Our main result says that, for natural choices of the pinning sequence $\{\epsilon_j\}$, localization (respectively delocalization) occurs if $\sigma^{-2}\sum_{ j\geq0}(j+1)\epsilon_j\geq\delta^{-1}$ (respectively $\le \delta$), for some universal $\delta < 1$. Our finding is reminiscent of the classical Bargmann-Jost-Pais criteria for the absence of bound states for the radial Schrödinger equation. The core of the proof is a recursive argument to bound the free energy of the model. Our approach is rather robust, which allows us to obtain similar results in the case where the random walk trajectory is replaced by a self-avoiding path $\gamma$ in $\mathbb Z^2$ with weight $\exp(-\beta |\gamma|)$, $|\gamma|$ being the length of the path and $\beta > 0$ a large enough parameter. This generalization is directly relevant for applications to the above mentioned contour models.

Article information

Electron. J. Probab. Volume 20 (2015), paper no. 8, 29 pp.

Accepted: 2 February 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 82C24: Interface problems; diffusion-limited aggregation

Random walks pinning entropic repulsion contour models

This work is licensed under a Creative Commons Attribution 3.0 License.


Caputo, Pietro; Martinelli, Fabio; Toninelli, Fabio. Multi-level pinning problems for random walks and self-avoiding lattice paths. Electron. J. Probab. 20 (2015), paper no. 8, 29 pp. doi:10.1214/EJP.v20-3849. https://projecteuclid.org/euclid.ejp/1465067114

Export citation


  • Bargmann, V. On the number of bound states in a central field of force. Proc. Nat. Acad. Sci. U. S. A. 38, (1952). 961–966.
  • Caputo, P.; Velenik, Y. A note on wetting transition for gradient fields. Stochastic Process. Appl. 87 (2000), no. 1, 107–113.
  • Pietro Caputo, Eyal Lubetzky, Fabio Martinelli, Allan Sly, and Fabio Lucio Toninelli, phScaling limit and cube-root fluctuations in sos surfaces above a wall, (2013), To appear on J. Eur. Math. Soc., preprint textttarXiv:1302.6941.
  • Caravenna, Francesco; Giacomin, Giambattista; Zambotti, Lorenzo. Sharp asymptotic behavior for wetting models in $(1+1)$-dimension. Electron. J. Probab. 11 (2006), no. 14, 345–362 (electronic).
  • Caravenna, Francesco; Pétrélis, Nicolas. A polymer in a multi-interface medium. Ann. Appl. Probab. 19 (2009), no. 5, 1803–1839.
  • Chayes, J. T.; Chayes, L. Ornstein-Zernike behavior for self-avoiding walks at all noncritical temperatures. Comm. Math. Phys. 105 (1986), no. 2, 221–238.
  • Deuschel, Jean-Dominique; Giacomin, Giambattista; Zambotti, Lorenzo. Scaling limits of equilibrium wetting models in $(1+1)$-dimension. Probab. Theory Related Fields 132 (2005), no. 4, 471–500.
  • Dobrushin, R.; Kotecky, R.; Shlosman, S. Wulff construction. A global shape from local interaction. Translated from the Russian by the authors. Translations of Mathematical Monographs, 104. American Mathematical Society, Providence, RI, 1992. x+204 pp. ISBN: 0-8218-4563-2
  • Doney, R. A. Local behaviour of first passage probabilities. Probab. Theory Related Fields 152 (2012), no. 3-4, 559–588.
  • Feller, William. An introduction to probability theory and its applications. Vol. II. Second edition John Wiley & Sons, Inc., New York-London-Sydney 1971 xxiv+669 pp.
  • Fisher, Michael E. Walks, walls, wetting, and melting. J. Statist. Phys. 34 (1984), no. 5-6, 667–729.
  • Giacomin, Giambattista. Random polymer models. Imperial College Press, London, 2007. xvi+242 pp. ISBN: 978-1-86094-786-5; 1-86094-786-7.
  • Ioffe, D. Ornstein-Zernike behaviour and analyticity of shapes for self-avoiding walks on ${\bf Z}^ d$. Markov Process. Related Fields 4 (1998), no. 3, 323–350.
  • Dima Ioffe, Senya Shlosman, and Fabio Lucio Toninelli, phInteraction versus entropic repulsion for low temperature ising polymers, J.Statist. Phys. in press (2015), textttarXiv:1407.3592.
  • Isozaki, Yasuki; Yoshida, Nobuo. Weakly pinned random walk on the wall: pathwise descriptions of the phase transition. Stochastic Process. Appl. 96 (2001), no. 2, 261–284.
  • Jost, R.; Pais, A. On the scattering of a particle by a static potential. Physical Rev. (2) 82, (1951). 840–851.
  • Sohier, Julien. The scaling limits of a heavy tailed Markov renewal process. Ann. Inst. Henri Poincaré Probab. Stat. 49 (2013), no. 2, 483–505.
  • Solomyak, Michael. On a class of spectral problems on the half-line and their applications to multi-dimensional problems. J. Spectr. Theory 3 (2013), no. 2, 215–235.