Random Riemannian Geometry in 4 Dimensions

Abstract: 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{8\pi^2}\bigg[\Delta^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*} d\mu(x):= e^{\gamma h(x)-\frac{\gamma^2}2k(x,x)}\,d \text{vol}g(x)\,, \end{equation*} and rigorously obtained a.s.~for $|\gamma|<\sqrt8$ 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.