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arXiv · 2502.06754

A switching identity for cable-graph loop soups and Gaussian free fields

Wendelin Werner
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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.

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Wendelin Werner. 2025-07-03. A switching identity for cable-graph loop soups and Gaussian free fields. https://arxiv.org/abs/2502.06754

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