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.)

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.)

By Louis-Pierre Arguin, Université de Montréal

The study of the distributions of extrema of a large collection of random variables dates back to the early 20th century and is well established in the case of independent or weakly correlated variables. Until recently, few sharp results were known in the case where the random variables are strongly correlated. In the last few years, there has been conceptual progress in describing the distribution of extrema of log-correlated Gaussian fields. This class of fields includes important examples such as branching Brownian motion and the 2D Gaussian free field. In this series of lectures, we will study the statistics of extrema of log-correlated Gaussian fields. The focus will be on explaining the guiding principles behind the results. We will use the example of branching Brownian motion to illustrate the method.

By Louis-Pierre Arguin, Université de Montréal

The study of the distributions of extrema of a large collection of random variables dates back to the early 20th century and is well established in the case of independent or weakly correlated variables. Until recently, few sharp results were known in the case where the random variables are strongly correlated. In the last few years, there has been conceptual progress in describing the distribution of extrema of log-correlated Gaussian fields. This class of fields includes important examples such as branching Brownian motion and the 2D Gaussian free field. In this series of lectures, we will study the statistics of extrema of log-correlated Gaussian fields. The focus will be on explaining the guiding principles behind the results. We will use the example of branching Brownian motion to illustrate the method.