Ursell functions in lattice gauge theory

TL;DR

The paper investigates Ursell functions in Ising lattice gauge theory, contrasting their sign behavior with the classic Ising model. It establishes that edge-based Ursell functions can be zero or change sign depending on n, parity, lattice dimension, and inverse temperature β, while Wilson-loop observables exhibit strictly positive first and second Ursell functions due to Griffiths inequalities. The main contribution is a low-temperature cluster-expansion framework that proves the existence of configurations of disjoint Wilson loops with positive U_n, highlighting a fundamentally different sign structure from the Ising case. The work leverages discrete exterior calculus, vortex-cluster analysis, and careful decomposition of Ursell functions to reveal how higher-order connected correlations behave in lattice gauge theories, with potential implications for understanding confinement-like phenomena via correlation structures.

Abstract

Ursell functions $U_n$ are higher-order generalizations of the covariance function, which capture the interactions between $n$ random variables. In the classical Ising model, as shown by Shlosman, when considering the spins at some locations, the sign of $U_{2n}$ alternates with $n$ and is independent of the locations of the spins considered. In this paper, we study the Ursell function in Ising lattice gauge theory. When the spins at the edges are used as random variables, we show that $U_n$ can be positive, negative, or zero depending on the configuration and the parameter $β$. When considering Wilson loops observables as random variables, using the tool of cluster expansion adapted to this setting, we prove that at sufficiently low temperature, for any number $n$ of disjoint Wilson loops, there exists a configuration of loops such that the Ursell function $U_n$ is positive. These results contrast sharply with the behavior observed for the Ising model.

Figures (18)

• Figure 1: A plaquette $p \in P(B_N)$ with it's four oriented boundary edges ${e_1,e_2,e_3,e_4 \in E(B_N)}$.
• Figure 2: A plaquette $p$ with its boundary on Figure (a) and coboundary on Figure (b) in the case $m=3$.
• Figure 3: The loop $\gamma$, s.t. $U_{10}^{\sigma}(W_\gamma) < 0$ for $\beta$ large.
• Figure 4: The loop $\gamma$ from Figure \ref{['fig:u10negative']} written as sums of two loops in three distinct ways.
• Figure 5: Example of a loop $\tilde{\gamma}$, s.t. $|\tilde{\gamma}|=12$ and $U_{12}^{\sigma}(W_{\tilde{\gamma}}) < 0$ at large $\beta$.

Theorems & Definitions (39)

• Theorem 1.1: Ursell function on edges
• Theorem 1.2: Ursell function on Wilson observables
• Remark 1.3
• Remark 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 3.1
• proof