Random Riemannian Geometry in 4 Dimensions

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+ $\mathsf{d}\mathsf{i}\mathsf{v}$\left(2$\mathsf{R}\mathsf{i}\mathsf{c}$-\frac23$\mathsf{s}\mathsf{c}\mathsf{a}\mathsf{l}$\right)\nabla \bigg]. \end{equation*} The kernel $k$ is invariant (modulo additive corrections) under conformal transformations, and it exhibits a precise logarithmic divergence $$\mid k(x,y)-\log \frac{1}{d(x,y)}\mid \le C\mathrm{.}$$ 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 $\mathrm{\mid }\gamma \mathrm{\mid }<\sqrt{8}$ as weak limit of the RHS with $h$ replaced by suitable regular approximations $(h_{\mathrm{\ell }}{)}_{\mathrm{\ell }\in \mathbb{N}}$. 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.