Spin systems from loop soups

Tim van de Brug, Federico Camia, Marcin Lis

Research output: Contribution to JournalArticleAcademicpeer-review

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 languageEnglish
Article number81
Pages (from-to)1-17
Number of pages17
JournalElectronic Journal of Probability
Volume23
DOIs
Publication statusPublished - 12 Sept 2018

Funding

TvdB and ML thank New York University Abu Dhabi for the hospitality during a visit in 2017. TvdB and FC thank the Indian Statistical Institute, Delhi Station for the hospitality during a visit in 2016. FC and ML thank Yves Le Jan for several interesting discussions. The authors also thank an anonymous referee for a careful reading of the manuscript and for providing useful comments. Part of the research was conducted while TvdB was at the Department of Mathematics of Vrije Universiteit Amsterdam. The research of ML was funded by EPSRC grants EP/I03372X/1 and EP/L018896/1.

FundersFunder number
New York University
Engineering and Physical Sciences Research CouncilEP/I03372X/1, EP/L018896/1
Vrije Universiteit Amsterdam

    Keywords

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

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