Variance asymptotics and scaling limits for Gaussian polytopes

Yukich, J., Lehigh University, USA

Consider the convex hull of n i.i.d. random variables with the standard normal distribution on d-dimensional Euclidean space. As n tends to infinity, we establish variance asymptotics for the volume and k-face functionals of the convex hull, resolving an open problem. Asymptotic variances and the scaling limit of the boundary of the convex hull are given in terms of functionals of germ-grain models having parabolic grains with apices at a thinned non-homogeneous Poisson point process. The scaling limit of the boundary of the convex hull coincides with that featuring in the geometric construction of the zero-viscosity solution of Burgers' equation. This is joint work with P. Calka.