Conformal Correlation Functions in the Brownian Loop Soup

Abstract

We define and study a set of operators that compute statistical properties of the Brownian Loop Soup, a conformally invariant gas of random Brownian loops (Brownian paths constrained to begin and end at the same point) in two dimensions. We prove that the correlation functions of these operators have many of the properties of conformal primaries in a conformal field theory, and compute their conformal dimension. The dimensions are real and positive, but have the novel feature that they vary continuously as a periodic function of a real parameter. We comment on the relation of the Brownian Loop Soup to the free field, and use this relation to establish that the central charge of the Loop Soup is twice its intensity.

25+5 pages, 3 figures. v2: minor changes v3: minor changes, comments added. arXiv admin note: substantial text overlap with arXiv:1501.04861

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• Two-Dimensional Critical Percolation: The Full Scaling Limit
• Eternal Inflation, Bubble Collisions, and the Disintegration of the Persistence of Memory
• SLE(6) and CLE(6) from Critical Percolation
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Cited by in corpus (9)

• Winding of simple walks on the square lattice
• Random walk loop soups and conformal loop ensembles
• Exact Correlation Functions in the Brownian Loop Soup
• Scalar conformal primary fields in the Brownian loop soup
• Brownian Loops and Conformal Fields
• The Brownian loop soup stress-energy tensor
• New Recipes for Brownian Loop Soups
• The expectation value of the number of loops and the left-passage probability in the double-dimer model
• The Brownian loop measure on Riemann surfaces and applications to length spectra