A Cellular Representation of the Potts Lattice Higgs Model

TL;DR

This paper introduces a novel cellular representation of the Potts lattice Higgs model, linking it to dependent percolation processes and establishing phase transition results in a high-dimensional lattice setting.

Contribution

It develops a new cellular representation of the Potts lattice Higgs model as dependent plaquette percolations and relates Wilson line expectations to topological events, proving phase transitions.

Findings

Representation of the model as dependent plaquette percolations

Expression of Wilson line expectations via topological events

Existence of phase transition in the model on  lattice

Abstract

The $i$-dimensional Potts lattice Higgs model is a random assignment of spins in $\mathbb{Z}_q$ to the $i$-dimensional cells of a cell complex induced by a Hamiltonian with a Potts interaction on the $(i+1)$-cells and an additional term playing the role of an external field. We develop a representation of this model as a pair of dependent plaquette percolations, and prove that Wilson line expectations can be expressed in terms of the probability of a topological event. As an application, we prove the existence of a phase transition for the Marcu--Fredenhagen ratio in the Potts lattice Higgs model on $\mathbb{Z}^d$ when $i=1.$