2012/10/30 by Li-Nan Chen, Dmitry Jakobson, Chen, Linan +1
Mathematics · Physics and Astronomy · #28C20 #31B30 #46E35 #60G57 #60G60 #Black Holes and Theoretical Physics #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Physics (math-ph) #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.1210.8051
openalex publication_date 2012/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This work aims to extend part of the two dimensional results of Duplantier and Sheffield on Liouville quantum gravity to four dimensions, and indicate possible extensions to other even-dimensional spaces R^(2n) as well as Riemannian manifolds. Let "Θ" be the Gaussian free field on R4 with the underlying Hilbert space being the Sobolev space H2 witb the inner product determined by the operator (I-Δ)2. Assume "θ" is a generic element from Θ. We consider a sequence of random Borel measures on R4, each of which is absolutely continuous with respect to the Lebesgue measure dx and the density function is given by the exponential of a centered Gaussian family parametrized by x in R4. We show that with probability 1, this sequence of measures weakly converges to a limit random measure which can be "formally" written as "exp(2γθ(x)dx". In this setting, we also prove a KPZ relation, which is the quadratic relation between the scaling exponent of a bounded Borel set on R4 under the Lebesgue measure and its counterpart under the random measure obtained above. Our approach is similar to the one used by Duplantier and Sheffield in 2D but with adaptations to R4.