First passage sets of the 2D continuum Gaussian free field

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]

Fichier principal

FPS1main.pdf (1.54 Mo)

ALE1.pdf (82.44 Ko)

ALE2.pdf (96.03 Ko)

ALEll.png (153.7 Ko)

BoundaryValues.pdf (38.57 Ko)

EXSHES.pdf (41.68 Ko)

FPSblue.png (102.51 Ko)

FPSilambda.png (51.87 Ko)

FPSlambda.png (51.13 Ko)

Flower.pdf (160.94 Ko)

Flower_cut.pdf (162.37 Ko)

LevelLines.pdf (34.19 Ko)

NotExactIdentity.pdf (35.83 Ko)

Reversibility_Level_Lines.pdf (35.18 Ko)

Reversibility_Level_Lines2.pdf (45.79 Ko)

TVSnConnected.pdf (61.63 Ko)

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https://hal.science/hal-01803800

Soumis le : mercredi 18 septembre 2019-07:47:04

Dernière modification le : jeudi 28 novembre 2024-15:42:26

Archivage à long terme le : samedi 8 février 2020-23:26:56

Dates et versions

hal-01803800 , version 1 (30-05-2018)

hal-01803800 , version 2 (18-09-2019)

Identifiants

HAL Id : hal-01803800 , version 2
ARXIV : 1706.07737
DOI : 10.1007/s00440-019-00941-1

Citer

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⟩

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