2023/11/28 by Dario, Paul, Garban, Christophe · 1 citation
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2311.16546
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>pc (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≥ 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 ℤd) or Poisson-Voronoi (in ℝd). 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.