Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Diffusion in planar Liouville quantum gravity

Nathanaël Berestycki

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Abstract

We construct the natural diffusion in the random geometry of planar Liouville quantum gravity. Formally, this is the Brownian motion in a domain $D$ of the complex plane for which the Riemannian metric tensor at a point $z\in D$ is given by $\exp(\gamma h(z))$, appropriately normalised. Here $h$ is an instance of the Gaussian free field on $D$ and $\gamma\in(0,2)$ is a parameter. We show that the process is almost surely continuous and enjoys certain conformal invariance properties. We also estimate the Hausdorff dimension of times that the diffusion spends in the thick points of the Gaussian free field, and show that it spends Lebesgue-almost all its time in the set of $\gamma$-thick points, almost surely. Similar but deeper results have been independently and simultaneously proved by Garban, Rhodes and Vargas.

Résumé

Nous construisons une diffusion naturelle associée ê la géométrie aléatoire de la gravité quantique de Liouville. Formellement, il s’agît d’un mouvement Brownien dans un domaine $D$ du plan complexe, muni d’un tenseur de Riemann donné par $\exp(\gamma h(z))$, correctement renomalisé. Ici $h$ est une réalisation du champ libre Gaussien sur $D$, et $\gamma\in\,]0,2[$ est un paramètre. Il est montré que ce processus est presque sûrement continu et possède certains propriétés d’invariance conforme. Une borne sur la dimension de Hausdorff des instants passés dans les points épais du champ libre Gaussien est obtenue, qui montre que cette diffusion passe Lebesue-presque tout son temps dans les points $\gamma$-épais, presque sûrement. Des résultats semblables mais plus profonds ont été indépendemment et simultanément obtenus par Garban, Rhodes et Vargas.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 3 (2015), 947-964.

Dates
Received: 20 February 2013
Revised: 17 December 2013
Accepted: 22 January 2014
First available in Project Euclid: 1 July 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1435759236

Digital Object Identifier
doi:10.1214/14-AIHP605

Mathematical Reviews number (MathSciNet)
MR3365969

Zentralblatt MATH identifier
1325.60125

Subjects
Primary: 60K37: Processes in random environments 60J67: Stochastic (Schramm-)Loewner evolution (SLE) 60J65: Brownian motion [See also 58J65]

Keywords
Brownian motion Gaussian free field Liouville quantum gravity Thick points

Citation

Berestycki, Nathanaël. Diffusion in planar Liouville quantum gravity. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 3, 947--964. doi:10.1214/14-AIHP605. https://projecteuclid.org/euclid.aihp/1435759236


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