### 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.

Original language | English (US) |
---|---|

Article number | 81 |

Journal | Electronic Journal of Probability |

Volume | 23 |

DOIs | |

State | Published - Jan 1 2018 |

### Fingerprint

### Keywords

- Brownian loop soup
- Conformal invariance
- Random field
- Random walk loop soup

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Electronic Journal of Probability*,

*23*, [81]. https://doi.org/10.1214/18-EJP200

**Spin systems from loop soups.** / van de Brug, Tim; Camia, Federico; Lis, Marcin.

Research output: Contribution to journal › Article

*Electronic Journal of Probability*, vol. 23, 81. https://doi.org/10.1214/18-EJP200

}

TY - JOUR

T1 - Spin systems from loop soups

AU - van de Brug, Tim

AU - Camia, Federico

AU - Lis, Marcin

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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.

AB - 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.

KW - Brownian loop soup

KW - Conformal invariance

KW - Random field

KW - Random walk loop soup

UR - http://www.scopus.com/inward/record.url?scp=85053460839&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85053460839&partnerID=8YFLogxK

U2 - 10.1214/18-EJP200

DO - 10.1214/18-EJP200

M3 - Article

AN - SCOPUS:85053460839

VL - 23

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

SN - 1083-6489

M1 - 81

ER -