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 \({\mathbb{R}^{2n}}\) as well as Riemannian manifolds. Let Θ be the Gaussian free field on \({\mathbb{R}^{4}}\) with the underlying Hilbert space \({H^{2}\left(\mathbb{R}^{4}\right)}\) and the inner product \({\left(\left(I-\Delta\right)^{2}\cdot,\cdot\right)_{L^{2}}}\), and θ a generic element from Θ. We consider a sequence of random Borel measures on \({\mathbb{R}^{4}}\), denoted by \({\left\{m_{\epsilon_{n}}^{\theta}\left({\rm d}x\right):n\geq1\right\}}\), 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 \({x \in \mathbb{R}^{4}}\). We show that with probability 1, \({m_{\epsilon_{n}}^{\theta}\left({\rm d}x\right)}\) weakly converges as \({\epsilon_{n} \downarrow 0}\), and the limit measure can be “formally” written as “\({m^{\theta}\left({\rm d}x\right) = e^{2\gamma\theta\left(x\right)}{\rm d}x}\) ”. In this setting, we also prove a KPZ relation, which is the quadratic relation between the scaling exponent of a bounded Borel set on \({\mathbb{R}^{4}}\) under the Lebesgue measure and its counterpart under the random measure \({m^{\theta}\left({\rm d}x\right)}\). Our approach is similar to the one used in Duplantier and Sheffield (Invent Math 185(2):333–393, 2011) with adaptations to \({\mathbb{R}^{4}}\).
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Communicated by Jens Marklof.
L. Chen was partly supported by FQRNT. D. Jakobson was partly supported by NSERC, FQRNT and Dawson Fellowships.
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Chen, L., Jakobson, D. Gaussian Free Fields and KPZ Relation in \({\mathbb{R}^4}\) . Ann. Henri Poincaré 15, 1245–1283 (2014). https://doi.org/10.1007/s00023-013-0277-1
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DOI: https://doi.org/10.1007/s00023-013-0277-1