First passage sets of the 2D continuum Gaussian free field

Juhan Aru (1, 2) , Titus Lupu (3, 4, 5) , Avelio Sepúlveda (6, 7)

1 ETH Zürich - Eidgenössische Technische Hochschule - Swiss Federal Institute of Technology [Zürich]
2 EPFL - Ecole Polytechnique Fédérale de Lausanne
3 CNRS - Centre National de la Recherche Scientifique
4 INSMI-CNRS - Institut National des Sciences Mathématiques et de leurs Interactions - CNRS Mathématiques
5 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
6 UCBL - Université Claude Bernard Lyon 1
7 ICJ - Institut Camille Jordan

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 Φ as a local set A so that Φ + 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 → | log(r)| 1/2 r 2 , by using Gaussian multiplicative chaos theory.

Mots clés: Gaussian free field; first passage sets; two-valued local sets; local set; Gaussian multiplicative chaos; Schramm-Loewner evolution

Domaines: Probabilités [math.PR]

Juhan Aru, Titus Lupu, Avelio Sepúlveda. First passage sets of the 2D continuum Gaussian free field. Probability Theory and Related Fields, 2020, 176 (3-4), pp.1303-1355. ⟨10.1007/s00440-019-00941-1⟩. ⟨hal-01803800v2⟩