Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On intermediate level sets of two-dimensional discrete Gaussian free field

Marek Biskup and Oren Louidor

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We consider the discrete Gaussian Free Field (DGFF) in scaled-up (square-lattice) versions of suitably regular continuum domains $D\subset\mathbb{C}$ and describe the scaling limit, including local structure, of the level sets at heights growing as a $\lambda$-multiple of the height of the absolute maximum, for any $\lambda\in(0,1)$. We prove that, in the scaling limit, the scaled spatial position of a typical point $x$ sampled from this level set is distributed according to a Liouville Quantum Gravity (LQG) measure in $D$ at parameter equal $\lambda$-times its critical value, the field value at $x$ has an exponential intensity measure and the configuration near $x$ reduced by the value at $x$ has the law of a pinned DGFF reduced by a suitable multiple of the potential kernel. In particular, the law of the total size of the level set, properly-normalized, converges to that of the total mass of the LQG measure. This sharpens considerably an earlier conclusion by Daviaud (Ann. Probab. 34 (2006) 962–986).


Nous considérons le champs Gaussien libre discret (DGFF) sur des versions renormalisées sur le réseau carré de domaines continus suffisamment réguliers $D\subset\mathbb{C}$ et décrivons la limite d’échelle, incluant la structure locale, des lignes de niveau lorsque que la hauteur croît comme $\lambda$-fois la hauteur du maximum absolu, pour tout $\lambda\in(0,1)$. Nous montons que, dans la limite d’échelle, la position normalisée d’un point typique $x$ tiré aléatoirement sur cette ligne de niveau a la loi de la mesure de Gravité Quantique de Liouville (LQG) dans $D$ avec paramètre égal à $\lambda$-fois sa valeur critique, la valeur du champs en $x$ ayant une mesure d’intensité exponentielle et la configuration près de $x$, réduite par la valeur en $x$, ayant la loi d’un champ libre épinglé DGFF réduit par un multiple adéquat du noyau potentiel. En particulier, la loi de la taille totale de la ligne de niveau, proprement normalisée, converge vers celle de la masse totale de la mesure LQG. Ceci améliore considérablement les résultats précédents de Daviaud (Ann. Probab. 34 (2006) 962–986).

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 4 (2019), 1948-1987.

Received: 15 December 2016
Revised: 17 May 2018
Accepted: 28 September 2018
First available in Project Euclid: 8 November 2019

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Mathematical Reviews number (MathSciNet)

Primary: 60G15: Gaussian processes 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 60G70: Extreme value theory; extremal processes 62G30: Order statistics; empirical distribution functions 60G55: Point processes 60G57: Random measures

Gaussian Free Field Level set Point process Liouville Quantum Gravity Scaling limit Conformal invariance


Biskup, Marek; Louidor, Oren. On intermediate level sets of two-dimensional discrete Gaussian free field. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 4, 1948--1987. doi:10.1214/18-AIHP939.

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