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- W76534522 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 $${left(left(I-Deltaright)^{2}cdot,cdotright)_{L^{2}}}$$ , and θ a generic element from Θ. We consider a sequence of random Borel measures on $${mathbb{R}^{4}}$$ , denoted by $${left{m_{epsilon_{n}}^{theta}left({rm d}xright):ngeq1right}}$$ , 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, $${m_{epsilon_{n}}^{theta}left({rm d}xright)}$$ weakly converges as $${epsilon_{n} downarrow 0}$$ , and the limit measure can be “formally” written as “ $${m^{theta}left({rm d}xright) = e^{2gammathetaleft(xright)}{rm d}x}$$ ”. 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 $${m^{theta}left({rm d}xright)}$$ . Our approach is similar to the one used in Duplantier and Sheffield (Invent Math 185(2):333–393, 2011) with adaptations to $${mathbb{R}^{4}}$$ ." @default.
- W76534522 created "2016-06-24" @default.
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- W76534522 date "2013-07-30" @default.
- W76534522 modified "2023-10-06" @default.
- W76534522 title "Gaussian Free Fields and KPZ Relation in $${mathbb{R}^4}$$ R 4" @default.
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- W76534522 doi "https://doi.org/10.1007/s00023-013-0277-1" @default.
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