The impact of disorder and non-convex interactions on delocalisation of height functions

TL;DR

This paper investigates how disorder and non-convex interactions influence phase transitions and delocalization in various height function models, extending previous results and establishing persistence of phase transitions under disorder.

Contribution

It extends known phase transition results to the Villain model and analyzes the effects of quenched disorder on dual height function models in two dimensions.

Findings

Phase transitions persist in the Villain model with disorder.

The roughening transition remains in 2D with certain quenched disorder.

Existence of a quantified rough phase for integer-valued height functions with Gaussian interactions.

Abstract

We study the behaviour of four spins systems (the XY model, the Villain model, the XY height function and the integer-valued Gaussian free field) in the presence of a non-elliptic quenched disorder. In the article [DG25], it was shown that the phase transitions of the XY model (the Berezinskii-Kosterlitz-Thouless phase transition in $d = 2$ and the order/disorder phase transition when $d \geq 3$) persist on the infinite cluster of a supercritical Bernoulli percolation. A first objective of this article is to extend these results to the Villain model. Our second objective is to analyse, for $d=2$, how the corresponding dual integer-valued height function models behave in the presence of a dual quenched disorder. These dual models are respectively the XY height function and the integer-valued Gaussian free field. Without disorder, these models are known to exhibit a phase transition in two dimensions called the roughening transition [FS81, Lam22b]. We show that this phase transition persists when the quenched disorder is given by enforcing $\varphi(x) = \varphi(y)$ independently with probability $\bar{p} < 1/2$ for neighboring sites $x, y$. Finally, we apply our methods to integer-valued height functions with annealed Gaussian interactions and prove the existence of a (quantified) rough phase. This includes all potentials of the form $|\nabla h|^p$ for $p \in (0, 2]$, recovering recent results of [OS25].