2024/01/23 by Karl‐Theodor Sturm, Sturm, Karl-Theodor · 1 citation
Mathematics · #Geometry and complex manifolds #Morphological variations and asymmetry #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2401.12676
We construct and analyze conformally invariant random fields on 4-dimensional Riemannian manifolds (M,g). These centered Gaussian fields h, called co-biharmonic Gaussian fields, are characterized by their covariance kernels k defined as the integral kernel for the inverse of the Paneitz operator \mathsf p=\frac18π2[Δ2+ div(2Ric-\frac23scal)∇ ]. The kernel k is invariant (modulo additive corrections) under conformal transformations, and it exhibits a precise logarithmic divergence |k(x,y)-log\frac1d(x,y)|≤ C. In terms of the co-biharmonic Gaussian field h, we define the quantum Liouville measure, a random measure on M, heuristically given as dμ(x):= e^γh(x)-\fracγ22k(x,x) d volg(x) , and rigorously obtained a.s.~for |γ|<√8 as weak limit of the RHS with h replaced by suitable regular approximations (h_ℓ)ℓ∈\mathbb N. For the flat torus M=\mathbb T4, 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.