First passage sets of the 2D continuum Gaussian free field

Abstract

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 on which the GFF does not go below $-a$. It 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 Minkowski 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. One of the highlights of this paper is identifying this measure as a Minkowski content measure in the non-integer gauge $r\mapsto |\log (r){|}^{1/2}{r}^2$, by using Gaussian multiplicative chaos theory.

The first version also contained arXiv:1805.09204, which is now a paper on its own; the third version is an all-around improved version ; 42 pages; 8 figures,

References in corpus (2)

• Critical Liouville measure as a limit of subcritical measures
• Global and Local Multiple SLEs for $\kappa \le 4$ and Connection Probabilities for Level Lines of GFF

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