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

TL;DR

This paper characterizes the threshold of the wetting transition for generalized pinning problems in random walks and self-avoiding paths, revealing conditions for localization and delocalization based on pinning potentials and variance.

Contribution

It provides a sharp criterion for the wetting transition in pinning models, extending results to self-avoiding paths and connecting to quantum mechanics criteria.

Findings

Localization occurs if $\sigma^{-2}\sum_{j ext{≥}0}(j+1)\epsilon_j extgreater ext{constant}$.

Delocalization occurs if $\sigma^{-2}\sum_{j ext{≥}0}(j+1)\epsilon_j extless ext{constant}$.

Results apply to both random walks and self-avoiding paths in lattice models.

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 Schr\"odinger 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.