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

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 Tarrès by means of a self-interacting process.

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