Year: 2008
Type: preprint
Abstract: Consider a bounded planar domain D, an instance h of the Gaussian free field on D (with Dirichlet energy normalized by 1/(2π)), 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 ε^{γ^2/2} e^{γh_ε(z)}dz, where dz is Lebesgue measure on D and h_ε(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.
Source: arXiv (Cornell University)
Authors: Duplantier, Bertrand, Sheffield, Scott
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funders: National Science Foundation, National Aeronautics and Space Administration, Microsoft Research, Centre de Recherches Mathématiques, Agence Nationale de la Recherche, Abdus Salam International Centre for Theoretical Physics, Centre National de la Recherche Scientifique, Université de Montréal, Office of International Science and Engineering
OpenAlex ID: W2949986008