Gaussian Free Fields and KPZ Relation in ℝ4

Abstract

This work aims to extend part of the two-dimensional results of Duplantier and Sheffield on Liouville quantum gravity (Invent Math 185(2):333-393, 2011) to four dimensions, and indicate possible extensions to other even-dimensional spaces ℝ2n as well as Riemannian manifolds. Let Θ be the Gaussian free field on ℝ4 with the underlying Hilbert space H2 (ℝ4) and the inner product ((I - Δ)2·,·)L2, and θ a generic element from Θ. We consider a sequence of random Borel measures on ℝ4, denoted by {m∈nθ(dx): n≥1}, 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 ∈ R4. We show that with probability 1, m∈nθ (dx) weakly converges as ∈n ↓ 0, and the limit measure can be "formally" written as "mθ(dx) = e2γθ(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 mθ (dx). Our approach is similar to the one used in Duplantier and Sheffield (Invent Math 185(2):333-393, 2011) with adaptations to ℝ4. © 2013 Springer Basel.