Juhan Aru : How to describe the 2D Gaussian free field

Abstract: 2D continuum Gaussian free field (GFF) is a canonical model for random surfaces. It has gained a central place due to its varied connections to SLE processes of Schramm, to Brownian loop soups and to what is sometimes called the Liouville measure. In these connections, the 2D GFF is often used as a tool - for example, it is used to explain the reversibility of SLE curves, and the Liouville measure is constructed as an exponential of the GFF. In this talk we will switch the focus and concentrate on the geometric and probabilistic properties of the GFF itself. We will highlight the usefulness of thinking of the 2D GFF as a generalization of Brownian motion, and see that even though the 2D GFF is not defined pointwise, one can talk of its level sets or study points that are above or below a certain height. The talk is based on joint works with T. Lupu, E. Powell, A. Sepúlveda and W. Werner.