Avelio Sepulveda: Excursion theory for the 2d Gaussian free field

Abstract: Traditional excursion theory is concerned with the decomposition of a 1-dimensional Markov processes in two parts: its zeros and its excursions, i.e., the path starting and ending on 0 that remain the same sign. We will explain how one can emulate the excursion theory for the 2d- Gaussian free field (GFF), the analogue of the Brownian motion when the time is replaced by a two-dimensional domain. The main technical difficulty is that the GFF is not a function but only a "generalised function" (Schwartz distribution), thus a priori it makes no sense to speak about its zeros, nor about its excursions. This talk is based on joint works with Juhan Aru, Titus Lupu and Wendelin Werner.