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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Abstract

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).

Résumé

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1573203620

Digital Object Identifier
doi:10.1214/18-AIHP939

Mathematical Reviews number (MathSciNet)
MR4029145

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.aihp/1573203620


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References

  • [1] Y. Abe. Maximum and minimum of local times for two-dimensional random walk. Electron. Commun. Probab. 20 (2015), paper no. 22, 14 pp.
  • [2] R. J. Adler. An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes. Institute of Mathematical Statistics Lecture Notes–Monograph Series 12, x+160 pp. Institute of Mathematical Statistics, Hayward, CA, 1990.
  • [3] J. Aru, E. Powell and A. Sepúlveda. Critical Liouville measure as a limit of subcritical measures, 2018. Available at arXiv:1802.08433.
  • [4] J. Barral and B. Mandelbrot. Non-degeneracy, moments, dimension, and multifractal analysis for random multiplicative measures (Random multiplicative multifractal measures, Part II). In Proc. Symp. Pures Math. 17–52, 72, Part II. AMS, Providence, RI, 2004.
  • [5] D. Belius and W. Wu. Maximum of the Ginzburg-Landau fields, 2016. Available at arXiv:1610.04195.
  • [6] N. Berestycki. An elementary approach to Gaussian multiplicative chaos. Electron. Commun. Probab. 22 (2017), paper no. 27, 12 pp.
  • [7] M. Biskup. Extrema of the two-dimensional Discrete Gaussian Free Field, 2017. Available at arXiv:1712.09972.
  • [8] M. Biskup, J. Ding and S. Goswami. Random walk in two-dimensional exponentiated Gaussian free field: recurrence and return probability, 2016. Available at arXiv:1611.03901.
  • [9] M. Biskup and O. Louidor. Conformal symmetries in the extremal process of two-dimensional discrete Gaussian Free Field, 2014. Available at arXiv:1410.4676.
  • [10] M. Biskup and O. Louidor. Extreme local extrema of two-dimensional discrete Gaussian free field. Comm. Math. Phys. 345 (1) (2016) 271–304.
  • [11] M. Biskup and O. Louidor. Full extremal process, cluster law and freezing for two-dimensional discrete Gaussian free field. Adv. Math. 330 (2018) 589–687.
  • [12] E. Bolthausen, J.-D. Deuschel and G. Giacomin. Entropic repulsion and the maximum of the two-dimensional harmonic crystal. Ann. Probab. 29 (4) (2001) 1670–1692.
  • [13] E. Bolthausen, J.-D. Deuschel and O. Zeitouni. Recursions and tightness for the maximum of the discrete, two dimensional Gaussian free field. Electron. Commun. Probab. 16 (2011) 114–119.
  • [14] M. Bramson, J. Ding and O. Zeitouni. Convergence in law of the maximum of the two-dimensional discrete Gaussian free field. Comm. Pure Appl. Math. 69 (1) (2016) 62–123.
  • [15] M. Bramson and O. Zeitouni. Tightness of the recentered maximum of the two-dimensional discrete Gaussian free field. Comm. Pure Appl. Math. 65 (2011) 1–20.
  • [16] S. Chatterjee, A. Dembo and J. Ding. On level sets of Gaussian fields, 2013. Available at arXiv:1310.5175.
  • [17] O. Daviaud. Extremes of the discrete two-dimensional Gaussian free field. Ann. Probab. 34 (2006) 962–986.
  • [18] B. Duplantier, R. Rhodes, S. Sheffield and V. Vargas. Renormalization of critical Gaussian multiplicative chaos and KPZ formula. Comm. Math. Phys. 330 (1) (2014) 283–330.
  • [19] B. Duplantier and S. Sheffield. Liouville quantum gravity and KPZ. Invent. Math. 185 (2) (2011) 333–393.
  • [20] X. Hu, J. Miller and Y. Peres. Thick point of the Gaussian free field. Ann. Probab. 38 (2) (2010) 896–926.
  • [21] J.-P. Kahane. Sur le chaos multiplicatif. Ann. Sci. Math. Québec 9 (1985) 105–150.
  • [22] T. Madaule. First order transition for the branching random walk at the critical parameter. Stoch. Proc. Appl. 126 (2) (2016) 470–502.
  • [23] J. Miller and S. Sheffield. Imaginary geometry IV: Interior rays, whole-plane reversibility, and space-filling trees, 2013. Available at arXiv:1302.4738.
  • [24] J. Miller and S. Sheffield. Liouville quantum gravity and the Brownian map I: The QLE(8/3, 0) metric, 2015. Available at arXiv:1507.00719.
  • [25] J. Miller and S. Sheffield. Imaginary geometry I: Interacting SLEs. Probab. Theory Related Fields 164 (3) (2016) 553–705.
  • [26] J. Miller and S. Sheffield. Imaginary geometry II: Reversibility of SLE$_{\kappa}(\rho_{1};\rho_{2})$ for $\kappa\in(0,4)$. Ann. Probab. 44 (3) (2016) 1647–1722.
  • [27] J. Miller and S. Sheffield. Imaginary geometry III: Reversibility of SLE$_{\kappa}$ for $\kappa\in(4,8)$. Ann. of Math. 184 (2) (2016) 455–486.
  • [28] J. Miller and S. Sheffield. Liouville quantum gravity and the Brownian map II: Geodesics and continuity of the embedding, 2016. Available at arXiv:1605.03563.
  • [29] R. Rhodes and V. Vargas. Gaussian multiplicative chaos and applications: A review. Probab. Surv. 11 (2014) 315–392.
  • [30] O. Schramm and S. Sheffield. Contour lines of the two-dimensional discrete Gaussian free field. Acta Math. 202 (1) (2009) 21–137.
  • [31] A. Shamov. On Gaussian multiplicative chaos. J. Funct. Anal. 270 (9) (2016) 3224–3261.
  • [32] S. Sheffield. Exploration trees and conformal loop ensembles. Duke Math. J. 147 (1) (2009) 79–129.
  • [33] S. Sheffield and W. Werner. Conformal loop ensembles: The Markovian characterization and the loop-soup construction. Ann. of Math. 176 (3) (2012) 1827–1917.