Level-sets of correlated Gaussian processes: connectivity and local geometry

Abstract: In this talk we will review some recent developments in the
study of percolation of Gaussian free field level-sets on the hypercubic
lattice in three and more dimensions. As a canonical percolation model
with slow algebraic decay of correlations, the new methodologies that
emerged in the course of these developments form a robust toolkit for
answering some classical questions in percolation theory in non-classical
contexts. We will spend considerable time discussing the motivation behind
this model, in particular its interplay with other very natural questions
in Probability theory as well the rich interplay with the potential theory
of random walks which is its another appealing feature. The main focus of
the talk will be a recent result on equality of several natural critical
parameters associated with this model --- relevant for understanding the
local geometry  of level-set components. No prior knowledge of percolation
theory is necessary. Based on joint work with H. Duminil-Copin, Pierre-F
Rodriguez and Franco Severo.