# Continuous interfaces with disorder: Even strong pinning is too weak in 2 dimensions

## TL;DR

This paper demonstrates that in two-dimensional continuous interface models, even strong pinning can be overcome by weak quenched disorder, leading to delocalization, contrasting with the behavior in higher dimensions.

## Contribution

It proves that weak quenched disorder causes delocalization in 2D interfaces despite strong pinning, extending understanding of disorder effects in statistical mechanics models.

## Key findings

- Weak disorder induces delocalization in 2D interfaces.
- Pinning becomes ineffective in 2D with any non-zero disorder.
- In higher dimensions, pinning dominates as force increases.

## Abstract

We consider statistical mechanics models of continuous height effective interfaces in the presence of a delta-pinning at height zero. There is a detailed mathematical understanding of the depinning transition in 2 dimensions without disorder. Then the variance of the interface height w.r.t. the Gibbs measure stays bounded uniformly in the volume for any positive pinning force and diverges like the logarithm of the pinning force when it tends to zero. How does the presence of a quenched disorder term in the Hamiltonian modify this transition? We show that an arbitarily weak random field term is enough to beat an arbitrarily strong delta-pinning in 2 dimensions and will cause delocalization. The proof is based on a rigorous lower bound for the overlap between local magnetizations and random fields in finite volume. In 2 dimensions it implies growth faster than the volume which is a contradiction to localization. We also derive a simple complementary inequality which shows that in higher dimensions the fraction of pinned sites converges to one when the pinning force tends to infinity.