Lattice gauge theories are probabilistic models from statistical mechanics, introduced in the 1970s as a discrete model of gauge theories from physics. From a probabilistic point of view, they can be seen as a higher-dimensional analogue of classical spin systems such as the Ising model: spins live on vertices and interact along edges in the Ising model; while spins live on edges and interact around plaquettes (2-cells) in lattice gauge theories.

In this talk, we will introduce the $\mathbb{Z}_2$ lattice gauge theory on the lattice $\mathbb{Z}^m, ~ m \geq 2$, and discuss some of its main questions, such as phase transitions and correlation inequalities.

We will then focus on Ursell functions, or connected correlation functions, which are higher-order analogues of covariance. While Shlosman's theorem (1986) shows that Ursell functions on spins in the Ising model have alternating signs, depending only on the number of spins considered, we will see that this picture breaks down for lattice gauge theory. In particular, at sufficiently low temperature and in dimension $m \geq 3$, one can prove that for any number of Wilson loops, there is a choice of Wilson loop observables whose Ursell function is positive.