First passage sets of the 2D continuum Gaussian free field

Juhan Aru, Titus Lupu, Avelio Sepúlveda

Published 2017-06-23Version 1

We introduce the first passage set (FPS) of constant level $-a$ of 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 or equal to $-a$, and is thus the two-dimensional analogue of the first hitting time of $-a$ by a one-dimensional Brownian motion. We provide an axiomatic characterization of the FPS, a continuum construction using level lines, and study its properties: it is a fractal set of zero Lebesgue measure and Hausdorff dimension 2 that is coupled with the GFF $\Phi$ as a local set $A$ so that $\Phi+a$ restricted to $A$ is a positive measure. We further prove that the metric graph FPS converges towards the continuum FPS. This allows us to make the informal description above rigorous, to show that the FPS can be represented as clusters of Brownian excursions and Brownian loops, and to revisit convergence results for level lines of the GFF.