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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2311.16546 |
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Table of Contents:
- 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.