Spin systems from loop soups

Tim van de Brug, Federico Camia, Marcin Lis

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Article number81
JournalElectronic Journal of Probability
StatePublished - 2018


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

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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