A Discovery Tour in Random Riemannian Geometry (2012.06796)

VIENNA PROBABILITY SEMINAR

In this talk, we discuss constructions and properties of massive Fractional Gaussian Fields h on a given Riemannian manifold (M,g) of bounded geometry. Our focus is on the regular case with Hurst parameter H > 0, the celebrated Liouville Geometry in 2d being borderline. We study random perturbations of the metric g by conformal factor the Fractional Gaussian Field h, provide estimates for basic geometric and functional-analytic objects relating to the random metric, such as intrinsic distance, spectral gap, and spectral bound, and we construct the random Brownian motion associated to the random geometry. The talk is based on joint work with Eva Kopfer and Karl-Theodor Sturm