Local metrics of the Gaussian free field
[Métriques locales du champ libre gaussien]
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.

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DOI : https://doi.org/10.5802/aif.3398
Classification : 60D05,  60G60
Mots clés : Champ libre gaussien, métriques locales, ensembles locaux, gravité quantique de Liouville
@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/}
}
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/

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