Ursell functions in lattice gauge theory

TL;DR

This paper investigates Ursell functions in lattice gauge theory, revealing that their sign behavior varies with configurations and parameters, especially showing positivity at low temperatures for Wilson loops, contrasting with classical Ising model results.

Contribution

It extends the study of Ursell functions to lattice gauge theory, demonstrating their sign variability and establishing conditions for positivity in Wilson loop observables at low temperatures.

Findings

Ursell functions can be positive, negative, or zero depending on configurations.

At low temperatures, there exist configurations where Ursell functions for Wilson loops are positive.

Results contrast with classical Ising model behavior.

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 $\beta$. 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.