The first passage sets of the 2D Gaussian free field: convergence and isomorphisms

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

1 ETH Zürich - Eidgenössische Technische Hochschule - Swiss Federal Institute of Technology [Zürich]
2 CNRS - Centre National de la Recherche Scientifique
3 INSMI-CNRS - Institut National des Sciences Mathématiques et de leurs Interactions - CNRS Mathématiques
4 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
5 PSPM - Probabilités, statistique, physique mathématique

In a previous article, we introduced 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 than or equal to −a. This description can be taken as a definition of the FPS for the metric graph GFF, and it justifies the analogy with the first hitting time of −a by a one-dimensional Brownian motion. In the current article, we prove that the metric graph FPS converges towards the continuum FPS in the Hausdorff metric. This allows us to show that the FPS of the continuum GFF can be represented as a union of clusters of Brownian excursions and Brownian loops, and to prove that Brownian loop soup clusters admit a non-trivial Minkowski content in the gauge r → | log r|^1/2 r^2. We also show that certain natural interfaces of the metric graph GFF converge to SLE4 processes.

Mots clés: Gaussian free field; Schramm-Loewner evolution; isomorphism theorems; loop-soup; local set; metric graph; conformal loop ensemble

Domaines: Probabilités [math.PR]

Juhan Aru, Titus Lupu, Avelio Sepúlveda. The first passage sets of the 2D Gaussian free field: convergence and isomorphisms. Communications in Mathematical Physics, 2020, 375 (3), pp.1885-1929. ⟨10.1007/s00220-020-03718-z⟩. ⟨hal-01798812⟩