{"title":"Phase Transitions for the XY Model in Non-uniformly Elliptic and Poisson-Voronoi Environments","authors":"Paul Dario, Christophe Garban","doi":"10.1007/s00220-025-05269-7","DOIUrl":null,"url":null,"abstract":"<div><p>The goal of this paper is to analyze how the celebrated phase transitions of the <i>XY</i> model are affected by the presence of a non-elliptic quenched disorder. In dimension <span>\\(d=2\\)</span>, we prove that if one considers an <i>XY</i> 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 <span>\\(p>p_c\\)</span> (site or edge). We also show that the <i>XY</i> model defined on a planar Poisson-Voronoi graph also undergoes a BKT phase transition. When <span>\\(d\\ge 3\\)</span>, we show in a similar fashion that the continuous symmetry breaking of the <i>XY</i> model at low enough temperature is not affected by the presence of quenched disorder such as supercritical percolation (in <span>\\(\\mathbb {Z}^d\\)</span>) or Poisson-Voronoi (in <span>\\(\\mathbb {R}^d\\)</span>). Adapting either Fröhlich–Spencer’s proof of existence of a BKT phase transition (Fröhlich and Spencer in Commun Math Phys 81(4):527–602, 1981) or the more recent proofs (Lammers in Probab Theory Relat Fields 182(1–2):531–550, 2022; van Engelenburg and Lis in Commun Math Phys 399(1):85–104, 2023; Aizenman et al. in Depinning in integer-restricted Gaussian Fields and BKT phases of two-component spin models, 2021. arXiv preprint arXiv:2110.09498; van Engelenburg and Lis in On the duality between height functions and continuous spin models, 2023. arXiv preprint arXiv:2303.08596) to such non-uniformly elliptic disorders appears to be non-trivial. Instead, our proofs rely on Wells’ correlation inequality (Wells in Some Moment Inequalities and a Result on Multivariable Unimodality. PhD thesis, Indiana University, 1977).</p></div>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":"406 5","pages":""},"PeriodicalIF":2.2000,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00220-025-05269-7.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/s00220-025-05269-7","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
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 \(d\ge 3\), 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 (Fröhlich and Spencer in Commun Math Phys 81(4):527–602, 1981) or the more recent proofs (Lammers in Probab Theory Relat Fields 182(1–2):531–550, 2022; van Engelenburg and Lis in Commun Math Phys 399(1):85–104, 2023; Aizenman et al. in Depinning in integer-restricted Gaussian Fields and BKT phases of two-component spin models, 2021. arXiv preprint arXiv:2110.09498; van Engelenburg and Lis in On the duality between height functions and continuous spin models, 2023. arXiv preprint arXiv:2303.08596) to such non-uniformly elliptic disorders appears to be non-trivial. Instead, our proofs rely on Wells’ correlation inequality (Wells in Some Moment Inequalities and a Result on Multivariable Unimodality. PhD thesis, Indiana University, 1977).
本文的目的是分析非椭圆淬火无序的存在如何影响XY模型的著名相变。在\(d=2\)维中,我们证明了在超临界渗流构型的无限簇上考虑XY模型,尽管存在淬灭失序,但仍会发生Berezinskii-Kosterlitz-Thouless (BKT)相变。证明适用于所有\(p>p_c\)(站点或边缘)。我们还证明了在平面泊松- voronoi图上定义的XY模型也经历了BKT相变。当\(d\ge 3\)时,我们以类似的方式显示,在足够低的温度下,XY模型的连续对称性破断不受超临界渗流(\(\mathbb {Z}^d\))或泊松-沃罗诺伊(\(\mathbb {R}^d\))等淬火无序存在的影响。采用Fröhlich-Spencer对BKT相变存在性的证明(Fröhlich and Spencer in commons Math Phys 81(4): 527-602, 1981)或更近期的证明(Lammers in Probab Theory relfields 182(1-2): 531-550, 2022;数学与物理学报(1):85-104,2009;Aizenman et al. in整数限制高斯场和双组分自旋模型的BKT相的脱钉,2021。arXiv预印arXiv:2110.09498;van Engelenburg和Lis,《论高度函数和连续自旋模型的对偶性》,2023。arXiv预印本arXiv:2303.08596)对这样的非均匀椭圆型紊乱似乎是非平凡的。相反,我们的证明依赖于Wells的相关不等式(Wells in Some Moment不等式和关于多变量单峰的结果)。博士论文,印第安纳大学,1977年)。
期刊介绍:
The mission of Communications in Mathematical Physics is to offer a high forum for works which are motivated by the vision and the challenges of modern physics and which at the same time meet the highest mathematical standards.