Loop soups at special intensities and the Gaussian free field

Abstract:

Discrete (resp. Brownian) loop soups are random collections of loops on graphs (resp. in the continuum) defined as Poisson point processes with the intensity measure given by what is known as the discrete (resp. Brownian) loop soup measure. In the discrete setting, we will explain special integrability properties loop soups obey when the intensity parameter of the Poisson point process has particular values. In particular, this will explain the relation to the discrete Gaussian free field (GFF) and random current representations of the Ising model. Moreover, we will explain the story in the continuum (Brownian) setting and explain analogous results as well as the relation of loop soup clusters to conformal loop ensembles (CLE).