Topological Phases in the Plaquette Random-Cluster Model and Potts Lattice Gauge Theory

Abstract

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, Fr\"olich, 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.