Free energy and quark potential in Ising lattice gauge theory via cluster expansion

We revisit the cluster expansion for Ising lattice gauge theory on ${\mathbb{Z}}^m,m\ge 3,$ with Wilson action, at a fixed inverse temperature $\beta$ in the low-temperature regime. We prove existence and analyticity of the infinite volume limit of the free energy and compute the first few terms in its expansion in powers of $e^{-\beta }$. We further analyze Wilson loop expectations and derive an estimate that shows how the lattice scale geometry of a loop is reflected in the large $\beta$ asymptotic expansion. Specializing to axis parallel rectangular loops ${\gamma }_{T,R}$ with side-lengths $T$ and $R$, we consider the limiting function $$V_{\beta }(R):=\underset{T\to \mathrm{\infty }}{\lim }-\frac{1}{T}\log ⟨W_{{\gamma }_{T,R}}{⟩}_{\beta },$$ known as the static quark potential in the physics literature. We verify existence of the limit (with an estimate on the convergence rate) and compute the first few terms in the expansion in powers of $e^{-\beta }$. As a consequence, a strong version of the perimeter law follows. We also treat $-\log ⟨W_{{\gamma }_{T,R}}{⟩}_{\beta }\mathrm{/}(T+R)$ as $T,R$ tend to infinity simultaneously and give analogous estimates.