Duncan Paul, Schweinhart Benjamin
Department of Mathematics, Indiana University, Bloomington, 47405 USA.
Department of Mathematical Sciences, George Mason University, Fairfax, 22030 VA USA.
Commun Math Phys. 2025;406(7):164. doi: 10.1007/s00220-025-05338-x. Epub 2025 Jun 4.
In 1983, Aizenman, Chayes, Chayes, Fröhlich, and Russo [1] proved that 2-dimensional Bernoulli plaquette percolation in exhibits a sharp phase transition for the event that a large rectangular loop is "bounded by a surface of plaquettes." We extend this result both to -dimensional plaquette percolation in and to a dependent model of plaquette percolation called the plaquette random-cluster model. As a consequence, we obtain a sharp phase transition for Wilson loop expectations in -dimensional -state Potts hyperlattice gauge theory on dual to that of the Potts model. Our proof is unconditional for Ising lattice gauge theory, but relies on a regularity conjecture for the random-cluster model in slabs when We also further develop the general theory of the -plaquette random cluster model and its relationship with -dimensional Potts lattice gauge theory.
1983年，艾森曼、蔡斯、蔡斯、弗勒利希和鲁索[1]证明，对于大矩形回路“由小块表面界定”这一事件，二维贝努利小块渗流呈现出尖锐的相变。我们将此结果推广到(\mathbb{Z}^d)中的(d)维小块渗流以及一种称为小块随机簇模型的相关小块渗流模型。因此，我们在与Potts模型对偶的(\mathbb{Z}^d)上的(q)态Potts超晶格规范理论中得到了威尔逊回路期望的尖锐相变。我们的证明对于伊辛晶格规范理论是无条件的，但当(d\geq3)时依赖于平板中随机簇模型的一个正则性猜想。我们还进一步发展了(q)小块随机簇模型的一般理论及其与(d)维Potts晶格规范理论的关系。
