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

Pietro Caputo,Fabio Toninelli,Fabio Martinelli

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 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.

Highlights

Introduction and motivationsConsider a one-dimensional integer-valued symmetric random walk starting at zero, conditioned to stay non-negative
If the walk has a reward ε > 0 for each return to zero, it is a classical fact that there exists a critical value εc such that for ε > εc the random walk has a positive density of returns to the origin while for ε < εc entropic repulsion prevails and the density of returns is zero; see e.g. [12] and references therein
In this work we consider a natural generalization where the pinning at the origin is replaced by a long range pinning potential ε = {εj}j 0, where εj 0 is the reward for a visit to the state j 0

Summary

Introduction and motivations

Consider a one-dimensional integer-valued symmetric random walk starting at zero, conditioned to stay non-negative. One of the motivations for this work stems from the mathematical analysis of contour models arising in low-temperature two-dimensional spin systems and related interface models In this context the random walk is replaced by a self-avoiding and weakly selfinteracting random lattice path with an effective diffusion constant σ2 ∼ e−β, where β is the inverse temperature. The analog of the pinning strength εj above typically decays like e−αβ(j+1) for j → ∞ with α > 1 Whether such long range potential is able to localize the contour is a key question in the analysis of large deviations problems such as e.g. the Wulff construction for the 2D Ising model [8] and for the (2 + 1)-dimensional Solid-on-Solid model [3]. The main idea of [13] consists in constructing, out of the self-interacting contour path, an effective random walk together with a renewal structure and prove delocalization for the latter

Models and results

Self-avoiding paths