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Spin systems from loop soups

Published version
Peer-reviewed

Type

Article

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Authors

van de Brug, T 
Camia, F 
Lis, M 

Abstract

We study spin systems defined by the winding of a random walk loop soup. For a particular choice of loop soup intensity, we show that the corresponding spin system is reflection-positive and is dual, in the Kramers-Wannier sense, to the spin system sgn(ϕ) where ϕ is a discrete Gaussian free field. In general, we show that the spin correlation functions have conformally covariant scaling limits corresponding to the one-parameter family of functions studied by Camia, Gandolfi and Kleban (Nuclear Physics B 902, 2016) and defined in terms of the winding of the Brownian loop soup. These functions have properties consistent with the behavior of correlation functions of conformal primaries in a conformal field theory. Here, we prove that they do correspond to correlation functions of continuum fields (random generalized functions) for values of the intensity of the Brownian loop soup that are not too large.

Description

Keywords

Brownian loop soup, random walk loop soup, conformal invariance, random field

Journal Title

Electronic Journal of Probability

Conference Name

Journal ISSN

1083-6489
1083-6489

Volume Title

23

Publisher

Institute of Mathematical Statistics
Sponsorship
Engineering and Physical Sciences Research Council (EP/I03372X/1)
Engineering and Physical Sciences Research Council (EP/L018896/1)
The research of ML was funded by EPSRC grants EP/I03372X/1 and EP/L018896/1.