A switching identity for cable-graph loop soups and Gaussian free fields
Abstract
We derive a "switching identity" that can be stated for critical Brownian loop-soups or for the Gaussian free field on a cable graph: It basically says that at the level of cluster configurations and at the more general level of the occupation time fields, conditioning two points on the cable-graph to belong to the same cluster of Brownian loops (or equivalently to the same sign-cluster of the GFF) amounts to adding a random odd number of independent Brownian excursions between these points to an otherwise unconditioned configuration. This explicit simple description of the conditional law of the clusters when a connection occurs has various direct consequences, in particular about the large scale behaviour of these sign-clusters on infinite graphs.
Cite
@article{arxiv.2502.06754,
title = {A switching identity for cable-graph loop soups and Gaussian free fields},
author = {Wendelin Werner},
journal= {arXiv preprint arXiv:2502.06754},
year = {2025}
}
Comments
Includes a new section with additional results about a switching property along loops and Lupu's intensity doubling conjecture in high dimensions