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

2025/02/10 by Wendelin Werner, Werner, Wendelin · 5 citations
Computer Science · #Algorithms and Data Compression #Data Management and Algorithms #Advanced Database Systems and Queries

paper · pdf · doi:10.48550/arxiv.2502.06754

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.

Cited by

• On cable-graph percolation between dimensions 2 and 3
• Singularity of the loops within a cable-graph loop-soup conditioned by its occupation time
• Intensity doubling for Brownian loop-soups in high dimensions
• On the gap between cluster dimensions of loop soups on ℝ³ and the metric graph of ℤ³
• On Loops in critical high-dimensional percolation

Related

• Singularity of the loops within a cable-graph loop-soup conditioned by its occupation time
• Intensity doubling for Brownian loop-soups in high dimensions
• Cluster explorations of the loop soup on a metric graph related to the Gaussian free field
• Arm events in critical planar loop soups
• Conditioning a Brownian loop-soup cluster on a portion of its boundary