Inverting the coupling of the signed Gausssian free field with a loop soup

TL;DR

This paper presents an inverse construction of a coupling between a signed Gaussian free field and a loop-soup, extending previous work to invert the known coupling and relate it to other models.

Contribution

It provides the first inverse construction starting from a signed GFF to generate a loop-soup, complementing Lupu's original forward coupling.

Findings

Constructed a loop-soup from a signed GFF using a self-interacting random walk

Inverted the coupling between the square of the GFF and the loop-soup

Deduced an inversion of the coupling between the random current and FK-Ising models

Abstract

Lupu introduced a coupling between a random walk loop-soup and a Gaussian free field, where the sign of the field is constant on each cluster of loops. This coupling is a signed version of isomorphism theorems relating the square of the GFF to the occupation field of Markovian trajectories. His construction starts with a loop-soup, and by adding additional randomness samples a GFF out of it. In this article we provide the inverse construction: starting from a signed free field and using a self-interacting random walk related to this field, we construct a random walk loop-soup. Our construction relies on the previous work by Sabot and Tarr\`es, which inverts the coupling from the square of the GFF rather than the signed GFF itself. As a consequence, we also deduce an inversion of the coupling between the random current and the FK-Ising random cluster models introduced by Lupu and Werner.