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Spin systems from loop soups. (English) Zbl 1400.82107
Summary: 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 \(\operatorname{sgn} (\varphi)\) 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 F. Camia et al. [Nucl. Phys., B 902, 483–507 (2016; Zbl 1332.82083)] 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.

MSC:
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
60G60 Random fields
60G18 Self-similar stochastic processes
60J65 Brownian motion
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
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