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

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 $φ(x) = φ(y)$ 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].