# Note on Bolthausen-Deuschel-Zeitouni's paper on the absence of a wetting transition for a pinned harmonic crystal in dimensions three and larger

## TL;DR

This paper revisits and corrects the proof of the absence of a wetting transition for a pinned harmonic crystal in three or more dimensions, extending previous results to more general pinning potentials.

## Contribution

It provides a corrected and generalized proof for the absence of a wetting transition in the case of square-potential pinning, removing reliance on an incorrect lower bound.

## Key findings

- Confirmed absence of wetting transition for harmonic crystals with square-pinning in dimensions three and higher.
- Provided a new proof method that does not depend on the disputed lower bound.
- Connected with recent alternative approaches by Giacomin and Lacoin.

## Abstract

The article [Bolthausen et al., 2000] provides a proof of the absence of a wetting transition for the discrete Gaussian free field conditioned to stay positive, and undergoing a weak delta-pinning at height 0. The proof is generalized to the case of a square pinning-potential replacing the delta-pinning, but it relies on a lower bound on the probability for the field to stay above the support of the potential, the proof of which appears to be incorrect. We provide a modified proof of the absence of a wetting transition in the square-potential case, which does not require the aforementioned lower bound. An alternative approach is given in a recent paper by Giacomin and Lacoin.