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.
Categories: math.PR math.DG math.MG