Brownian Loops and Conformal Fields

Abstract: The main topic of these lecture notes is the continuum scaling limit of planar lattice models. One reason why this topic occupies an important place in the theory of probability and mathematical statistical physics is that scaling limits provide the link between statistical mechanics and (Euclidean) field theory. In order to explain the main ideas behind the concept of scaling limit, I will focus on a "toy" model that exhibits the typical behavior of statistical mechanical models at and near the critical point. This model, known as the random walk loop soup, is actually interesting in its own right. It can be described as a Poissonian ensemble of lattice loops, or a lattice gas of loops since it fits within the ideal gas framework of statistical mechanics. After introducing the model and discussing some interesting connections with the discrete Gaussian free field, I will present some results concerning its scaling limit, which leads to a Poissonian ensemble of continuum loops known as the Brownian loop soup. The latter was introduced by Lawler and Werner and is a very interesting object with connections to the Schramm-Loewner Evolution and various models of statistical mechanics. In the second part of the lectures, I will use the Brownian loop soup to construct a family of functions that behave like correlation functions of a conformal field. I will then use these functions and their derivation to introduce the concept of conformal field and to explore the connection between scaling limits and conformal fields.