## Electronic Journal of Probability

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

#### Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067114

Digital Object Identifier
doi:10.1214/EJP.v20-3849

Mathematical Reviews number (MathSciNet)
MR3311221

Zentralblatt MATH identifier
1327.60183

Rights

#### Citation

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

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