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

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Bibliographic Details
Main Author:	Werner, Wendelin
Format:	Preprint
Published:	2025
Subjects:	Probability Mathematical Physics
Online Access:	Acceder al recurso
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author	Werner, Wendelin
author_facet	Werner, Wendelin
contents	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.
format	Preprint
id	arxiv_https___arxiv_org_abs_2502_06754
institution	arXiv
publishDate	2025
record_format	arxiv
spellingShingle	A switching identity for cable-graph loop soups and Gaussian free fields Werner, Wendelin Probability Mathematical Physics 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.
title	A switching identity for cable-graph loop soups and Gaussian free fields
topic	Probability Mathematical Physics
url	https://arxiv.org/abs/2502.06754

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