Spin systems from loop soups

Marcin Lis,Tim Van De Brug,Federico Camia

doi:10.1214/18-ejp200

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.

Highlights

The random walk loop soup (RWLS) was introduced by Lawler and Trujillo Ferreras [28] as a discrete analog of the Brownian loop soup (BLS) of Lawler and Werner [29]
For sufficiently low intensities λ, the intersecting loops form clusters whose outer boundaries are distributed like Conformal Loop Ensembles (CLEs) [43, 44]
CLEs are the unique ensembles of planar, non-crossing and non-self-crossing loops satisfying a natural conformal restriction property that is conjecturally satisfied by the continuum scaling limits of interfaces in two-dimensional models of statistical physics

Summary

Introduction

The random walk loop soup (RWLS) was introduced by Lawler and Trujillo Ferreras [28] as a discrete analog of the Brownian loop soup (BLS) of Lawler and Werner [29]. Motivated by the work of Freivogel and Kleban [17] on bubble nucleation in theories of eternal inflation, Camia, Gandolfi and Kleban defined and computed certain statistical correlation functions that characterize aspects of the BLS distribution [6] They looked in particular at the net winding of all the loops around a given set of points and found results consistent with the behavior of correlation functions of primary fields in a conformal field theory (CFT). We show that, for values of the intensity λ of the BLS that are not too large, but still including the most interesting case, λ = 1/2, one can construct continuum Euclidean fields (random generalized functions) whose correlation functions are the functions obtained in [6] These fields do not seem to correspond to any currently known CFT. As pointed out in [6], the putative CFT associated with those correlation functions has the interesting feature that the conformal dimensions of the primary operators are real and positive, but vary continuously and are periodic functions of a real parameter

RWLS spin fields and the discrete Gaussian free field

Griffiths inequalities and reflection positivity

Convergence of correlation functions

Brownian loop soup winding fields