Conditioning two cable-graph points to lie in the same Brownian loop-soup cluster adds an odd-numbered Poisson cloud of Brownian excursions between them, yielding an exact law for the conditional cluster and its GFF analogue.
On the number of incipient spanning clusters
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A switching identity for cable-graph loop soups and Gaussian free fields
Conditioning two cable-graph points to lie in the same Brownian loop-soup cluster adds an odd-numbered Poisson cloud of Brownian excursions between them, yielding an exact law for the conditional cluster and its GFF analogue.