Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Level lines of the Gaussian free field with general boundary data

Ellen Powell and Hao Wu

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We study the level lines of a Gaussian free field in a planar domain with general boundary data $F$. We show that the level lines exist as continuous curves under the assumption that $F$ is regulated (i.e., admits finite left and right limits at every point), and satisfies certain inequalities. Moreover, these level lines are a.s. determined by the field. This allows us to define and study a generalization of the $\operatorname{SLE}_{4}(\underline{\rho})$ process, now with a continuum of force points. A crucial ingredient is a monotonicity property in terms of the boundary data which strengthens a result of Miller and Sheffield and is also of independent interest.


Nous étudions les lignes de niveau d’un champ libre Gaussien dans un domaine $D$ du plan, avec condition au bord générale donnée par une fonction $F$. Nous montrons que ces lignes existent comme courbes continues sous l’hypothèse que $F$ est une fonction réglée (i.e., $F$ admet une limite à droite et à gauche en tous points) et satisfait certaines inégalités. De plus, ces lignes de niveau sont presque sûrement determinées par le champ. Cela nous permet de définir et d’étudier une généralisation des courbes $\operatorname{SLE}_{4}(\underline{\rho})$ avec un continuum de points marqués. Un ingrédient essentiel de la preuve est une propriété de monotonicité en termes de la condition au bord, d’un intérêt indépendant, qui améliore un théorème de Miller et Sheffield.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 4 (2017), 2229-2259.

Received: 29 September 2015
Revised: 22 August 2016
Accepted: 22 August 2016
First available in Project Euclid: 27 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Gaussian free field Level lines Schramm Loewner evolution


Powell, Ellen; Wu, Hao. Level lines of the Gaussian free field with general boundary data. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 4, 2229--2259. doi:10.1214/16-AIHP789.

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