We consider level sets of the Gaussian free field on the $d$-dimensional lattice, for $d>2$, above a given real-valued height $h$. This defines a percolation model with strong, algebraically decaying correlations. We prove a conjecture of Lebowitz asserting that the sign clusters of this field, i.e. the level sets above height $h=0$, contain a unique infinite connected component. As a central ingredient, we exploit a certain algebraic correspondence which relates the free field to a Poissonian soup of bi-infinite random walk trajectories known as random interlacements, originally introduced by Sznitman to study local limits of random walks on large, asymptotically transient graphs. The interlacement trajectories can be fruitfully used to build large clusters. Representations of similar flavor can be traced back to celebrated works of Symanzik and Brydges-Fröhlich-Spencer.

Talk I will set the stage and provide an overview of the argument. Talk II will fill in certain technical details, including state-of-the-art methods to deal with percolation models in the presence of strong correlations, and the extension of these techniques to a so-called "cable system", obtained by joining adjacent vertices on the lattice by "cables", which provides a handy continuous geometric structure. Based on joint work with Alexander Drewitz and Alexis Prévost.

# Percolation of sign clusters for the Gaussian free field I

Pierre-Francois Rodriguez

University of California

May 10, 2018