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2018 Spin systems from loop soups
Tim van de Brug, Federico Camia, Marcin Lis
Electron. J. Probab. 23: 1-17 (2018). DOI: 10.1214/18-EJP200


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 $\mathrm{sgn} (\varphi )$ where $\varphi $ 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.


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Tim van de Brug. Federico Camia. Marcin Lis. "Spin systems from loop soups." Electron. J. Probab. 23 1 - 17, 2018.


Received: 26 March 2018; Accepted: 15 July 2018; Published: 2018
First available in Project Euclid: 12 September 2018

zbMATH: 1400.82107
MathSciNet: MR3858909
Digital Object Identifier: 10.1214/18-EJP200

Primary: 60G18, 60G60, 82B20, 82B41


Vol.23 • 2018
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