Liouville Quantum Gravity and KPZ
Abstract
Consider a bounded planar domain D, an instance h of the Gaussian free field on D (with Dirichlet energy normalized by 1/(2\pi)), and a constant 0 < gamma < 2. The Liouville quantum gravity measure on D is the weak limit as epsilon tends to 0 of the measures \epsilon^{\gamma^2/2} e^{\gamma h_\epsilon(z)}dz, where dz is Lebesgue measure on D and h_\epsilon(z) denotes the mean value of h on the circle of radius epsilon centered at z. Given a random (or deterministic) subset X of D one can define the scaling dimension of X using either Lebesgue measure or this random measure. We derive a general quadratic relation between these two dimensions, which we view as a probabilistic formulation of the KPZ relation from conformal field theory. We also present a boundary analog of KPZ (for subsets of the boundary of D). We discuss the connection between discrete and continuum quantum gravity and provide a framework for understanding Euclidean scaling exponents via quantum gravity.
 Publication:

arXiv eprints
 Pub Date:
 August 2008
 arXiv:
 arXiv:0808.1560
 Bibcode:
 2008arXiv0808.1560D
 Keywords:

 Mathematics  Probability;
 Mathematical Physics
 EPrint:
 56 pages. Revised version contains more details. To appear in Inventiones