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.