A switching identity for cable-graph loop soups and Gaussian free fields
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915370654236672 |
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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 |