Ornstein-Zernike decay of Wilson line observables in the free phase of the ${\mathbb{Z}}_2$ lattice Higgs model

In the physics literature, the Wilson line observable is believed to have a phase transition between a region with pure exponential decay and a region with Ornstein-Zernike type corrections. In~\cite{f2024b}, we confirmed the first part of this prediction. In this paper, we complement these results by showing that if $\kappa$ is small and $\beta$ large compared to the length of the line, then Wilson line expectations have exponential decay with Ornstein-Zernike type behavior.