Spin systems from loop soups

TL;DR

This paper explores the connection between spin systems derived from loop soups and conformal field theory, establishing dualities, reflection positivity, and conformally covariant scaling limits for certain intensities.

Contribution

It demonstrates that specific spin systems from loop soups are dual to Gaussian free fields and have conformally covariant limits, linking probabilistic models to conformal field theory.

Findings

Spin systems are reflection-positive at certain intensities.

Duality with Gaussian free fields is established.

Conformally covariant scaling limits are identified for specific parameters.

Abstract

We study spin systems defined by the winding of a random walk loop soup. For a particular choice of loop soup intensity, we show that the corresponding spin system is reflection-positive and is dual, in the Kramers-Wannier sense, to the spin system $\text{sgn}(\varphi)$ where $\varphi$ is a discrete Gaussian free field. In general, we show that the spin correlation functions have conformally covariant scaling limits corresponding to the one-parameter family of functions studied by Camia, Gandolfi and Kleban (Nuclear Physics B, 902, 2016) and defined in terms of the winding of the Brownian loop soup. These functions have properties consistent with the behavior of correlation functions of conformal primaries in a conformal field theory. Here, we prove that they do correspond to correlation functions of continuum fields (random generalized functions) for values of the intensity of the Brownian loop soup that are not too large.