An equivalence between gauge-twisted and topologically conditioned scalar Gaussian free fields
Résumé
We study on the metric graphs two types of scalar Gaussian free fields (GFF), the usual one and the one twisted by a {−1, 1}-valued gauge field. We show that the latter can be obtained, up to an additional deterministic transformation, by conditioning the first on a topological event. This event is that all the sign clusters of the field should be trivial for the gauge field, that is to say should not contain loops with holonomy −1. We also express the probability of this topological event as a ratio of two determinants of Laplacians to the power 1/2, the usual Laplacian and the gauge-twisted Laplacian. As an example, this gives on annular planar domains the probability that no sign cluster of the metric graph GFF surrounds the inner hole of the domain.
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