Cluster explorations of the loop soup on a metric graph related to the Gaussian free field

TL;DR

This paper investigates the properties of the loop soup on a metric graph, establishing relations with the Gaussian free field, and introduces a domain Markov property and conditioned laws using Fleming–Viot processes.

Contribution

It introduces new relations between conditioned loop soups, a domain Markov property, and connects local times to Gaussian free fields, extending previous discrete models to metric graphs.

Findings

Established a relation between conditioned loop soups and local times.

Proved a domain Markov property for the loop soup.

Derived the law of the loop soup conditioned on occupation field.

Abstract

We consider the loop soup at intensity ${1\over 2}$ conditioned on having local time $0$ on a set of vertices with positive occupation field in their vicinities. We give a relation between this loop soup and the usual loop soup conditioned on its local times. We deduce a domain Markov property for the loop soup, in the vein of the discrete Markov property proved by Werner: when exploring a cluster, the bridges outside the cluster form a Poisson point process. We show how it is related to the property due to Le Jan that the local times of the loop soup are distributed as the squares of a Gaussian free field. Finally, our results naturally give the law of the loop soup conditioned on its occupation field via Fleming--Viot processes. The discrete analog of this question was addressed by Werner in terms of the random current model, and by Lupu, Sabot and Tarr\`es by means of a self-interacting process.