no. 5
Local metrics of the Gaussian free field
[Métriques locales du champ libre gaussien]
Gwynne, Ewain1 ; Miller, Jason1
1 Department of Mathematics Faculty of Mathematics, Wilberforce Road, Cambridge CB3 0WA (United Kingdom)
Annales de l'Institut Fourier, Tome 70 (2020) no. 5, pp. 2049-2075.

Nous introduisons la notion de métrique locale d’un champ libre gaussien h. Il s’agit d’une propriété d’ine distance aléatoire couplée avec h d’une maniére locale qui rappelle la notion d’ensembles locaux du champ libre gaussien. Nous établissons des critéres pour vérifier que deux métriques locales associées á un même champ libre gaussien sont Lipschitz-équivalentes, ou pour vérifier qu’une métrique locale est en fait une fonction détereministe du champ libre.

Ces résultats sont utilisés dans des travaux ultérieurs qui établissent l’existence, l’unicité et d’autres propriétés de la métrique associée á la gravité quantique de Liouville pour tout paramétre γ∈(0,2), mais le présent article ne requiert aucune connaissance sur la gravité quantique.

We introduce the concept of a local metric of the Gaussian free field (GFF) h, which is a random metric coupled with h in such a way that it depends locally on h in a certain sense. This definition is a metric analog of the concept of a local set for h. We establish general criteria for two local metrics of the same GFF h to be bi-Lipschitz equivalent to each other and for a local metric to be a.s. determined by h. Our results are used in subsequent works which prove the existence, uniqueness, and basic properties of the γ-Liouville quantum gravity (LQG) metric for all γ∈(0,2), but no knowledge of LQG is needed to understand this paper.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3398
Classification : 60D05, 60G60
Keywords: Gaussian free field, local metrics, local sets, Liouville quantum gravity
Mots-clés : Champ libre gaussien, métriques locales, ensembles locaux, gravité quantique de Liouville

Gwynne, Ewain 1 ; Miller, Jason 1

1 Department of Mathematics Faculty of Mathematics, Wilberforce Road, Cambridge CB3 0WA (United Kingdom)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AIF_2020__70_5_2049_0,
     author = {Gwynne, Ewain and Miller, Jason},
     title = {Local metrics of the {Gaussian} free field},
     journal = {Annales de l'Institut Fourier},
     pages = {2049--2075},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {70},
     number = {5},
     year = {2020},
     doi = {10.5802/aif.3398},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3398/}
}
TY  - JOUR
AU  - Gwynne, Ewain
AU  - Miller, Jason
TI  - Local metrics of the Gaussian free field
JO  - Annales de l'Institut Fourier
PY  - 2020
SP  - 2049
EP  - 2075
VL  - 70
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3398/
DO  - 10.5802/aif.3398
LA  - en
ID  - AIF_2020__70_5_2049_0
ER  - 
%0 Journal Article
%A Gwynne, Ewain
%A Miller, Jason
%T Local metrics of the Gaussian free field
%J Annales de l'Institut Fourier
%D 2020
%P 2049-2075
%V 70
%N 5
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3398/
%R 10.5802/aif.3398
%G en
%F AIF_2020__70_5_2049_0
Gwynne, Ewain; Miller, Jason. Local metrics of the Gaussian free field. Annales de l'Institut Fourier, Tome 70 (2020) no. 5, pp. 2049-2075. doi : 10.5802/aif.3398. https://aif.centre-mersenne.org/articles/10.5802/aif.3398/

[1] Ding, Jian; Dubédat, Julien; Dunlap, Alexander; Falconet, Hugo Tightness of Liouville first passage percolation for γ∈(0,2) (2019) (https://arxiv.org/abs/1904.08021) | Zbl

[2] Dubédat, Julien; Falconet, Hugo; Gwynne, Ewain; Pfeffer, Joshua; Sun, Xin Weak LQG metrics and Liouville first passage percolation, Probab. Theory Relat. Fields, Volume 178 (2020) no. 1-2, pp. 369-436 | DOI | MR | Zbl

[3] Duplantier, Bertrand; Sheffield, Scott Liouville quantum gravity and KPZ, Invent. Math., Volume 185 (2011) no. 2, pp. 333-393 | DOI | MR

[4] Efron, Bradley; Stein, Charles M. The jackknife estimate of variance, Ann. Stat., Volume 9 (1981) no. 3, pp. 586-596 | MR

[5] Gwynne, Ewain; Miller, Jason Conformal covariance of the Liouville quantum gravity metric for γ∈(0,2) (2019) (https://arxiv.org/abs/1905.00384, to appear in Ann. Inst. Henri Poincaré)

[6] Gwynne, Ewain; Miller, Jason Existence and uniqueness of the Liouville quantum gravity metric for γ∈(0,2) (2019) (https://arxiv.org/abs/1905.00383, to appear in Invent. Math.)

[7] Gwynne, Ewain; Miller, Jason Confluence of geodesics in Liouville quantum gravity for γ∈(0,2), Ann. Probab., Volume 48 (2020) no. 4, pp. 1861-1901 | DOI | MR

[8] Gwynne, Ewain; Miller, Jason; Sheffield, Scott Harmonic functions on mated-CRT maps, Electron. J. Probab., Volume 24 (2019), 58, 55 pages | MR | Zbl

[9] Kahane, Jean-Pierre Sur le chaos multiplicatif, Ann. Sci. Math. Qué., Volume 9 (1985) no. 2, pp. 105-150 | MR

[10] Le Gall, Jean-François Uniqueness and universality of the Brownian map, Ann. Probab., Volume 41 (2013) no. 4, pp. 2880-2960 | DOI | MR

[11] Miermont, Grégory The Brownian map is the scaling limit of uniform random plane quadrangulations, Acta Math., Volume 210 (2013) no. 2, pp. 319-401 | DOI | MR

[12] Miller, Jason; Qian, Wei The geodesics in Liouville quantum gravity are not Schramm–Loewner evolutions, Probab. Theory Relat. Fields, Volume 177 (2020) no. 3-4, pp. 677-709 | DOI | MR | Zbl

[13] Miller, Jason; Sheffield, Scott Imaginary geometry I: Interacting SLEs, Probab. Theory Relat. Fields, Volume 164 (2016) no. 3-4, pp. 553-705 | DOI | MR | Zbl

[14] Miller, Jason; Sheffield, Scott Imaginary geometry II: Reversibility of SLEκ(ρ1;ρ2) for κ∈(0,4), Ann. Probab., Volume 44 (2016) no. 3, pp. 1647-1722 | DOI | MR

[15] Miller, Jason; Sheffield, Scott Imaginary geometry III: Reversibility of SLEκ for κ∈(4,8), Ann. Math., Volume 184 (2016) no. 2, pp. 455-486 | DOI | MR | Zbl

[16] Miller, Jason; Sheffield, Scott Liouville quantum gravity and the Brownian map II: geodesics and continuity of the embedding (2016) (https://arxiv.org/abs/1605.03563)

[17] Miller, Jason; Sheffield, Scott Liouville quantum gravity and the Brownian map III: the conformal structure is determined (2016) (https://arxiv.org/abs/1608.05391, to appear in Probab. Theory Relat. Fields) | Zbl

[18] Miller, Jason; Sheffield, Scott Quantum Loewner evolution, Duke Math. J., Volume 165 (2016) no. 17, pp. 3241-3378 | DOI | MR | Zbl

[19] Miller, Jason; Sheffield, Scott Imaginary geometry IV: Interior rays, whole-plane reversibility, and space-filling trees, Probab. Theory Relat. Fields, Volume 169 (2017) no. 3-4, pp. 729-869 | DOI | MR | Zbl

[20] Miller, Jason; Sheffield, Scott Liouville quantum gravity and the Brownian map I: The QLE(8/3,0) metric, Invent. Math., Volume 219 (2020) no. 1, pp. 75-152 | DOI | MR | Zbl

[21] Rhodes, Rémi; Vargas, Vincent Gaussian multiplicative chaos and applications: A review, Probab. Surv., Volume 11 (2014), pp. 315-392 | DOI | MR | Zbl

[22] Schramm, Oded; Sheffield, Scott A contour line of the continuum Gaussian free field, Probab. Theory Relat. Fields, Volume 157 (2013) no. 1-2, pp. 47-80 | DOI | MR | Zbl

[23] Sheffield, Scott Gaussian free fields for mathematicians, Probab. Theory Relat. Fields, Volume 139 (2007) no. 3-4, pp. 521-541 | DOI | MR

[24] Sheffield, Scott Conformal weldings of random surfaces: SLE and the quantum gravity zipper, Ann. Probab., Volume 44 (2016) no. 5, pp. 3474-3545 | DOI | MR

Cité par Sources :