Wilson lines in the lattice Higgs model at strong coupling

We consider the 4D fixed length lattice Higgs model with Wilson action for the gauge field and structure group ${\mathbb{Z}}_n$. We study Wilson line observables in the strong coupling regime and compute their asymptotic behavior with error estimates. Our analysis is based on a high-temperature representation of the lattice Higgs measure combined with Poisson approximation. We also give a short proof of the folklore result that Wilson line (and loop) expectations exhibit perimeter law decay whenever the Higgs field coupling constant is positive.