论文标题
Plaquette随机群集模型和Potts晶格理论中的拓扑阶段
Topological Phases in the Plaquette Random-Cluster Model and Potts Lattice Gauge Theory
论文作者
论文摘要
有限立方体上的$ i $二维Plaquette随机群集模型是$ i $ plaquettes的随机复合体，每种配置的概率与$$ p^{\ plaquettes}}}}}（1-p）（1-p）^{ b} _ {i-1}}，$$其中$ q \ geq 1 $是一个真实参数，$ \ mathbf {b} _ {i-1} $表示$（i-1）$ - 同源群的等级，该组在指定的系数字段中具有系数。当$ q $是PRIME，并且系数字段为$ \ Mathbb {f} _Q $时，此型号与$（i-1）$ -Dimensional $ q $ -State Potts Potts lattice Gauge理论相结合。我们证明，在$ \ mathbb {z}^d $中$（i-1）$ - 循环的概率在Plaquette Random-Cluster模型中是无原样的，等于相应的通用Wilson Loop可变的期望。这为Aizenman，Chayes，Chayes，Frölich和Russo的主张提供了第一个严格的理由，即Wilson Loop变量与循环在plaquettes相互作用的系统中受到表面界定的事件之间存在确切的关系。我们还证明，$ i $ dimensional Plaquette随机群集模型在$ 2i $ - 维圆环上表现出在自偶点$ p _ {\ mathrm {sd}}} \ Mathrel {\ Mathrel {\ vcenter {\ vcenter {\ vcenter {：}} = =：}} = =： \ frac {\ sqrt {q}} {1+ \ sqrt {q}} $在同源渗透的意义上。这意味着从局部到非本地行为的广义Swendsen的质量变化。
The $i$-dimensional plaquette random-cluster model on a finite cubical complex is the random complex of $i$-plaquettes with each configuration having probability proportional to $$p^{\text{# of plaquettes}}(1-p)^{\text{# of complementary plaquettes}}q^{\mathbf{ b}_{i-1}},$$ where $q\geq 1$ is a real parameter and $\mathbf{b}_{i-1}$ denotes the rank of the $(i-1)$-homology group with coefficients in a specified coefficient field. When $q$ is prime and the coefficient field is $\mathbb{F}_q$, this model is coupled with the $(i-1)$-dimensional $q$-state Potts lattice gauge theory. We prove that the probability that an $(i-1)$-cycle in $\mathbb{Z}^d$ is null-homologous in the plaquette random-cluster model equals the expectation of the corresponding generalized Wilson loop variable. This provides the first rigorous justification for a claim of Aizenman, Chayes, Chayes, Frölich, and Russo that there is an exact relationship between Wilson loop variables and the event that a loop is bounded by a surface in an interacting system of plaquettes. We also prove that the $i$-dimensional plaquette random-cluster model on the $2i$-dimensional torus exhibits a sharp phase transition at the self-dual point $p_{\mathrm{sd}} \mathrel{\vcenter{:}}= \frac{\sqrt{q}}{1+\sqrt{q}}$ in the sense of homological percolation. This implies a qualitative change in the generalized Swendsen--Wang dynamics from local to non-local behavior.
