Spectral dimension of Liouville quantum gravity

Rémi Rhodes (1) , Vincent Vargas (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

This paper is concerned with computing the spectral dimension of 2d-Liouville quantum gravity. As a warm-up, we first treat the simple case of boundary Liouville quantum gravity. We prove that the spectral dimension is 1 via an exact expression for the boundary Liouville Brownian motion and heat kernel. Then we treat the 2d-case via a decomposition of time integral transforms of the Liouville heat kernel into Gaussian multiplicative chaos of Brownian bridges. We show that the spectral dimension is 2 in this case, as announced by physicists (see Ambj\o rn and al. in \cite{amb}) fifteen years ago.

Mots clés: Liouville quantum gravity; Liouville Brownian motion; Gaussian multiplicative chaos; Liouville heat kernel; spectral dimension

Domaines: Probabilités [math.PR]; Physique mathématique [math-ph]

Rémi Rhodes, Vincent Vargas. Spectral dimension of Liouville quantum gravity. Annales de l'Institut Henri Poincaré, 2014, 15. ⟨hal-00819451⟩