The First Passage Sets of the 2D Gaussian Free Field: Convergence and Isomorphisms

In a previous article, we introduced the first passage set (FPS) of constant level -aof the two-dimensional continuum Gaussian free field (GFF) on finitely connected domains. Informally, it is the set of points in the domain that can be connected to the boundary by a path along which the GFF is greater than or equal to -aThis description can be taken as a definition of the FPS for the metric graph GFF, and in the current article, we prove that the metric graph FPS converges towards the continuum FPS in the Hausdorff distance. We also draw numerous consequences; in particular, we obtain a relatively simple proof of the fact that certain natural interfaces of the metric graph GFF converge to SLE4 level lines. These results improve our understanding of the continuum GFF, by strengthening its relationship with the critical Brownian loop-soup. Indeed, a new construction of the FPS using clusters of Brownian loops and excursions helps to strengthen the known GFF isomorphism theorems, and allows us to use Brownian loop-soup techniques to prove technical results on the geometry of the GFF. We also obtain a new representation of Brownian loop-soup clusters, and as a consequence, we prove that the clusters of a critical Brownian loop-soup admit a non-trivial Minkowski content in the gauge r & x21a6;|logr|1/2r2.