Current expansion and couplings for Ising lattice gauge theory

TL;DR

This paper develops a surface-based expansion and couplings for Ising lattice gauge theory, providing new proofs and insights into Wilson loop expectations, correlation decay, and inequalities across all temperature regimes.

Contribution

It introduces a random current expansion and switching lemma for Ising lattice gauge theory, connecting it with other representations and deriving key properties.

Findings

Wilson loop expectation is non-negative at all eta

Exponential decay of correlations at small and large eta

Wilson loop expectations increase with eta

Abstract

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 $\beta$.