We introduce the concept of a local metric of the Gaussian free field (GFF) , which is a random metric coupled with in such a way that it depends locally on in a certain sense. This definition is a metric analog of the concept of a local set for . We establish general criteria for two local metrics of the same GFF to be bi-Lipschitz equivalent to each other and for a local metric to be a.s. determined by . Our results are used in subsequent works which prove the existence, uniqueness, and basic properties of the -Liouville quantum gravity (LQG) metric for all , but no knowledge of LQG is needed to understand this paper.
Nous introduisons la notion de métrique locale d’un champ libre gaussien . Il s’agit d’une propriété d’ine distance aléatoire couplée avec 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 , mais le présent article ne requiert aucune connaissance sur la gravité quantique.
Revised:
Accepted:
Published online:
Keywords: Gaussian free field, local metrics, local sets, Liouville quantum gravity
Mot clés : Champ libre gaussien, métriques locales, ensembles locaux, gravité quantique de Liouville
Gwynne, Ewain 1; Miller, Jason 1
@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, Volume 70 (2020) no. 5, pp. 2049-2075. doi : 10.5802/aif.3398. https://aif.centre-mersenne.org/articles/10.5802/aif.3398/
[1] Tightness of Liouville first passage percolation for (2019) (https://arxiv.org/abs/1904.08021) | Zbl
[2] 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] Liouville quantum gravity and KPZ, Invent. Math., Volume 185 (2011) no. 2, pp. 333-393 | DOI | MR
[4] The jackknife estimate of variance, Ann. Stat., Volume 9 (1981) no. 3, pp. 586-596 | MR
[5] Conformal covariance of the Liouville quantum gravity metric for (2019) (https://arxiv.org/abs/1905.00384, to appear in Ann. Inst. Henri Poincaré)
[6] Existence and uniqueness of the Liouville quantum gravity metric for (2019) (https://arxiv.org/abs/1905.00383, to appear in Invent. Math.)
[7] Confluence of geodesics in Liouville quantum gravity for , Ann. Probab., Volume 48 (2020) no. 4, pp. 1861-1901 | DOI | MR
[8] Harmonic functions on mated-CRT maps, Electron. J. Probab., Volume 24 (2019), 58, 55 pages | MR | Zbl
[9] Sur le chaos multiplicatif, Ann. Sci. Math. Qué., Volume 9 (1985) no. 2, pp. 105-150 | MR
[10] Uniqueness and universality of the Brownian map, Ann. Probab., Volume 41 (2013) no. 4, pp. 2880-2960 | DOI | MR
[11] 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] 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] Imaginary geometry I: Interacting SLEs, Probab. Theory Relat. Fields, Volume 164 (2016) no. 3-4, pp. 553-705 | DOI | MR | Zbl
[14] Imaginary geometry II: Reversibility of for , Ann. Probab., Volume 44 (2016) no. 3, pp. 1647-1722 | DOI | MR
[15] Imaginary geometry III: Reversibility of for , Ann. Math., Volume 184 (2016) no. 2, pp. 455-486 | DOI | MR | Zbl
[16] Liouville quantum gravity and the Brownian map II: geodesics and continuity of the embedding (2016) (https://arxiv.org/abs/1605.03563)
[17] 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] Quantum Loewner evolution, Duke Math. J., Volume 165 (2016) no. 17, pp. 3241-3378 | DOI | MR | Zbl
[19] 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] 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] Gaussian multiplicative chaos and applications: A review, Probab. Surv., Volume 11 (2014), pp. 315-392 | DOI | MR | Zbl
[22] 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] Gaussian free fields for mathematicians, Probab. Theory Relat. Fields, Volume 139 (2007) no. 3-4, pp. 521-541 | DOI | MR
[24] Conformal weldings of random surfaces: SLE and the quantum gravity zipper, Ann. Probab., Volume 44 (2016) no. 5, pp. 3474-3545 | DOI | MR
Cited by Sources: