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 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.
Type de document :
Pré-publication, Document de travail
2018
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https://hal.archives-ouvertes.fr/hal-01803800
Contributeur : Titus Lupu <>
Soumis le : mercredi 30 mai 2018 - 22:57:03
Dernière modification le : lundi 18 mars 2019 - 16:00:00
Document(s) archivé(s) le : vendredi 31 août 2018 - 19:14:10

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  • HAL Id : hal-01803800, version 1

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Juhan Aru, Titus Lupu, Avelio Sepúlveda. First passage sets of the 2D continuum Gaussian free field. 2018. 〈hal-01803800〉

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