Current expansion and couplings for Ising lattice gauge theory

In this note, we discuss a random current expansion and a switching lemma for Ising lattice gauge theory at all choices of inverse temperature $\beta$, leading to summation over surfaces. We also describe couplings of this expansion with other representations, including the high-temperature expansion and the cluster expansion. We deduce some simple consequences, including several expressions for the Wilson loop expectation (at any $\beta$), a new proof of the area law estimate for sufficiently small $\beta$, and a proof of exponential decay of correlations for small and large $\beta.$ We also derive a few results analogous to corresponding results for the Ising model. In particular, we show that the Wilson loop expectation is non-negative at any $\beta$ and give an alternative short proof of Griffith's second inequality and, as a consequence, show that the Wilson loop expectations are increasing in $\beta$ for all $\beta$.