Random Riemannian Geometry in 4 Dimensions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913205557657600 |
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| author | Sturm, Karl-Theodor |
| author_facet | Sturm, Karl-Theodor |
| contents | We construct and analyze conformally invariant random fields on 4-dimensional Riemannian manifolds $(M,g)$. These centered Gaussian fields $h$, called \emph{co-biharmonic Gaussian fields}, are characterized by their covariance kernels $k$ defined as the integral kernel for the inverse of the \emph{Paneitz operator} \begin{equation*}\mathsf p=\frac1{8π^2}\bigg[Δ^2+
\mathsf{div}\left(2\mathsf{Ric}-\frac23\mathsf{scal}\right)\nabla \bigg]. \end{equation*} The kernel $k$ is invariant (modulo additive corrections) under conformal transformations, and it exhibits a precise logarithmic divergence $$\Big|k(x,y)-\log\frac1{d(x,y)}\Big|\le C.$$ In terms of the co-biharmonic Gaussian field $h$, we define the \emph{quantum Liouville measure}, a random measure on $M$, heuristically given as \begin{equation*}
dμ(x):= e^{γh(x)-\frac{γ^2}2k(x,x)}\,d \text{vol}_g(x)\,, \end{equation*} and rigorously obtained a.s.~for $|γ|<\sqrt8$ as weak limit of the RHS with $h$ replaced by suitable regular approximations $(h_\ell)_{\ell\in\mathbb N}$.
For the flat torus $M=\mathbb T^4$, we provide discrete approximations of the Gaussian field and of the Liouville measures in terms of semi-discrete random objects, based on Gaussian random variables on the discrete torus and piecewise constant functions in the isotropic Haar system. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_12676 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Random Riemannian Geometry in 4 Dimensions Sturm, Karl-Theodor Probability Differential Geometry Metric Geometry 60G15, 58J65, 31C25 We construct and analyze conformally invariant random fields on 4-dimensional Riemannian manifolds $(M,g)$. These centered Gaussian fields $h$, called \emph{co-biharmonic Gaussian fields}, are characterized by their covariance kernels $k$ defined as the integral kernel for the inverse of the \emph{Paneitz operator} \begin{equation*}\mathsf p=\frac1{8π^2}\bigg[Δ^2+ \mathsf{div}\left(2\mathsf{Ric}-\frac23\mathsf{scal}\right)\nabla \bigg]. \end{equation*} The kernel $k$ is invariant (modulo additive corrections) under conformal transformations, and it exhibits a precise logarithmic divergence $$\Big|k(x,y)-\log\frac1{d(x,y)}\Big|\le C.$$ In terms of the co-biharmonic Gaussian field $h$, we define the \emph{quantum Liouville measure}, a random measure on $M$, heuristically given as \begin{equation*} dμ(x):= e^{γh(x)-\frac{γ^2}2k(x,x)}\,d \text{vol}_g(x)\,, \end{equation*} and rigorously obtained a.s.~for $|γ|<\sqrt8$ as weak limit of the RHS with $h$ replaced by suitable regular approximations $(h_\ell)_{\ell\in\mathbb N}$. For the flat torus $M=\mathbb T^4$, we provide discrete approximations of the Gaussian field and of the Liouville measures in terms of semi-discrete random objects, based on Gaussian random variables on the discrete torus and piecewise constant functions in the isotropic Haar system. |
| title | Random Riemannian Geometry in 4 Dimensions |
| topic | Probability Differential Geometry Metric Geometry 60G15, 58J65, 31C25 |
| url | https://arxiv.org/abs/2401.12676 |