Phase transitions for the $XY$ model in non-uniformly elliptic and Poisson-Voronoi environments

TL;DR

The paper establishes that the XY model’s canonical phase transitions are robust to non-uniformly elliptic quenched disorder from percolation and Poisson-Voronoi environments. By leveraging Wells' inequality to compare quenched and annealed versions, and using Ginibre’s correlation inequality alongside a renormalization scheme on supercritical clusters, the authors prove a BKT transition in $d=2$ for percolation and Voronoi settings, and long-range order in $d\ge3$ at sufficiently large inverse temperature. The results extend to high-density percolation and generalized extended lattices, providing almost-sure statements and detailed percolation-renormalization arguments. This work deepens understanding of how geometric randomness in the underlying graph affects continuous-symmetry spin systems and offers techniques potentially applicable to related models such as the Heisenberg or Villain variants. The findings have significance for statistical physics in disordered media, including implications for Polyakov-type conjectures and the study of quenched randomness in low-temperature phases.

Abstract

The goal of this paper is to analyze how the celebrated phase transitions of the $XY$ model are affected by the presence of a non-elliptic quenched disorder. In dimension $d=2$, we prove that if one considers an $XY$ model on the infinite cluster of a supercritical percolation configuration, the Berezinskii-Kosterlitz-Thouless (BKT) phase transition still occurs despite the presence of quenched disorder. The proof works for all $p>p_c$ (site or edge). We also show that the $XY$ model defined on a planar Poisson-Voronoi graph also undergoes a BKT phase transition. When $d\geq 3$, we show in a similar fashion that the continuous symmetry breaking of the $XY$ model at low enough temperature is not affected by the presence of quenched disorder such as supercritical percolation (in $\mathbb{Z}^d$) or Poisson-Voronoi (in $\mathbb{R}^d$). Adapting either Fröhlich-Spencer's proof of existence of a BKT phase transition or the more recent proofs of Lammers, van Engelenburg-Lis and Aizenman-Harel-Peled-Shapiro to such non-uniformly elliptic disorders appears to be non-trivial. Instead, our proofs rely on a relatively little known correlation inequality called Wells' inequality.

Figures (16)

• Figure 1.1: The two types of quenched disorders for which we prove that the BKT phase is non-trivial: on the left is pictured a supercritical edge-percolation configuration $\omega_p$ (with $p=0.66> p_{c, \mathrm{edge}}(2)=\tfrac{1}{2}$). On the right, the sites represent a Poisson point process $P$ in the plane $\mathbb{R}^2$. Each point $x\in P$ carries a spin $\sigma_x\in \mathbf{S}^1$. Two such spins interact if and only if their Voronoi cells (pictured with black edges) intersect. Or equivalently, if they are connected by an edge in the dual Delaunay triangulation (pictured in green).
• Figure 2.1: A realization of the $XY$ model on $\mathbb{Z}^2$
• Figure 2.2: An example of a pre-good box for site (top) and edge (bottom) percolation. In both cases, the cluster $\mathcal{C}(\Lambda)$ is drawn in blue.
• Figure 2.3: The graph $\mathbb{Z}^d$ and the extended graph $\mathbb{Z}^d_n$ (with $n = 3$).
• Figure 2.4: A realization of the $XY$ model on the extended lattice $\mathbb{Z}^d_n$ (with $d = 2$ and $n = 3$) with heterogeneous temperatures. The two inverse temperatures $\beta_1$ and $\beta_2$ are displayed in red and blue respectively

Theorems & Definitions (81)

• Theorem 1.1: Phase transitions for the $XY$ model on a high density Bernoulli site percolation cluster
• Remark 1
• Corollary 1.2: Phase transitions for the $XY$ model on a high density Bernoulli edge percolation cluster
• Theorem 1.3: Phase transitions for the $XY$ model on a supercritical Bernoulli percolation cluster
• Remark 2
• Corollary 1.4
• Theorem 1.5: Phase transitions for the $XY$ model on Poisson-Voronoi
• Remark 3
• Definition 2.1: $XY$ model with general coupling constants
• Definition 2.2: $XY$ model with annealed disorder