Bertrand Duplantier: Liouville quantum gravity and KPZ
Bertrand Duplantier, Institute for Theoretical Physics, Saclay (France)
Tid: On 2012-05-30 kl 13.15
Plats: Room 3721, Lindstedtsvägen 25, 7th floor, Department of mathematics, KTH
In Liouville quantum gravity, the (properly regularized) quantum area measure is defined in terms of the exponential of the Gaussian free field (GFF). It is expected to describe the scaling limit of random planar maps after uniformization.
We present a (mathematically rigorous) probabilistic and geometrical proof of the Knizhnik-Polyakov-Zamolodchikov (KPZ) relation between fractal scaling exponents in a Euclidean domain and in Liouville quantum gravity. The proof extends to the boundary geometry. A duality relation further exists between high and low strengths of the GFF, which mirrors that of the Schramm-Loewner evolution (SLE).
Joint work with Scott Sheffield (MIT).