KPZ formula derived from Liouville heat kernel

Abstract

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.

22 pages

References in corpus (4)

• The topological structure of scaling limits of large planar maps
• KPZ in one dimensional random geometry of multiplicative cascades
• Liouville heat kernel: regularity and bounds
• Quantum gravity and the KPZ formula

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