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

TL;DR

This paper introduces a switching identity for Brownian loop-soups and Gaussian free fields on cable graphs, providing a new explicit description of cluster conditioning and its implications for large-scale behavior.

Contribution

It establishes a novel switching identity linking cluster conditioning to adding Brownian excursions, enhancing understanding of loop-soups and GFFs on cable graphs.

Findings

Explicit switching identity for cluster conditioning.

Connection between loop-soups and Gaussian free fields.

Implications for large-scale cluster behavior on infinite graphs.

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.