KPZ formula derived from Liouville heat kernel

Nathanael Berestycki (1) , Christophe Garban (2, 3) , Rémi Rhodes (4) , Vincent Vargas (5)

1 DPMMS - Department of Pure Mathematics and Mathematical Statistics
2 ICJ - Institut Camille Jordan
3 PSPM - Probabilités, statistique, physique mathématique
4 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
5 DMA - Département de Mathématiques et Applications - ENS Paris

In this paper, we establish the Knizhnik--Polyakov--Zamolodchikov (KPZ) formula of Liouville quantum gravity, using the heat kernel of Liouville Brownian motion. This derivation of the KPZ formula was first suggested by F. David and M. Bauer in order to get a geometrically more intrinsic way of measuring the dimension of sets in Liouville quantum gravity. We also provide a careful study of the (no)-doubling behaviour of the Liouville measures in the appendix, which is of independent interest.

Mots clés: Gaussian multiplicative chaos; Liouville measures; Liouville quantum gravity; heat kernel; KPZ formula

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

Nathanael Berestycki, Christophe Garban, Rémi Rhodes, Vincent Vargas. KPZ formula derived from Liouville heat kernel. Journal of the London Mathematical Society, 2016, 94 (1), pp.186-208. ⟨hal-01071004⟩