# Mini-course 2, Session 2: Brownian Loops and Conformal Fields

By Federico Camia, Vrije Universiteit Amsterdam

Poissonian ensembles of Brownian loops and their discrete (lattice) counterpart have attracted considerable attention in recent years, particularly because of their conformal invariance and connections to the Schramm-Loewner Evolution (SLE) and the Gaussian free field. They are often called "loop soups" and fit into the "ideal gas" framework of statistical mechanics. I will first introduce the random walk loop soup and discuss some connections with the discrete Gaussian free field. I will then present some results about the convergence of the random walk loop soup to the Brownian loop soup and explain the relevance of the latter in connection with SLE and statistical mechanics. Finally, I will define a set of functionals of the Brownian loop soup whose correlation functions behave like "conformal primaries" in a conformal field theory (i.e., they scale covariantly under conformal maps). Similar functionals were first introduced in an attempt to formulate a conformal field theory of "eternal inflation", a cosmological theory that attempts to explain the origin of our universe. (Partly based on joint work with Tim van de Brug and Marcin Lis, and with Alberto Gandolfi and Matthew Kleban. No prior knowledge of conformal field theory or cosmology is required.)