Weighted graphs and complex Gaussian free fields

TL;DR

This paper establishes a new proof connecting loop measures and complex Gaussian fields through a combinatorial lemma about directed currents in a complex loop soup, enhancing understanding of their relationship.

Contribution

It introduces a novel combinatorial approach to prove the isomorphism between loop measures and complex Gaussian fields, providing fresh insights into their connection.

Findings

Proves a combinatorial lemma about directed currents in a complex loop soup.

Provides a new proof of the isomorphism between loop measures and complex Gaussian fields.

Enhances theoretical understanding of the relationship between loop soups and Gaussian fields.

Abstract

We prove a combinatorial lemma about the distribution of directed currents in a complex "loop soup" and use it to give a new proof of the isomorphism relating loop measures and complex Gaussian fields.