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2015 Multi-level pinning problems for random walks and self-avoiding lattice paths
Pietro Caputo, Fabio Martinelli, Fabio Toninelli
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Electron. J. Probab. 20: 1-29 (2015). DOI: 10.1214/EJP.v20-3849

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.

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Pietro Caputo. Fabio Martinelli. Fabio Toninelli. "Multi-level pinning problems for random walks and self-avoiding lattice paths." Electron. J. Probab. 20 1 - 29, 2015. https://doi.org/10.1214/EJP.v20-3849

Information

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

zbMATH: 1327.60183
MathSciNet: MR3311221
Digital Object Identifier: 10.1214/EJP.v20-3849

Subjects:
Primary: 60K35
Secondary: 82B41, 82C24

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