Variance asymptotics and scaling limits for Gaussian polytopes

Abstract

Let K n be the convex hull of i.i.d. random variables distributed according to the standard normal distribution on R d . We establish variance asymptotics as n → ∞ for the re-scaled intrinsic volumes and k-face functionals of K n , k ∈ {0, 1, ..., d − 1}, resolving an open problem [27]. Variance asymptotics are given in terms of functionals of germ-grain models having parabolic grains with apices at a Poisson point process on R d−1 × R with intensity e h dhdv. The scaling limit of the boundary of K n as n → ∞ converges to a festoon of parabolic surfaces, coinciding with that featuring in the geometric construction of the zero viscosity solution to Burgers' equation with random input.American Mathematical Society 2010 subject classifications. Primary 60F05, 52A20; Secondary 60D05, 52A23

Gaussian polytopes: A cumulant-based approach

“…While the rescaled geometric functionals satisfy a weak spatial localization property in the upper half-space, such a behaviour is no longer true in the global set-up. This remarkable but unavoidable phenomenon, explained in detail in [10] and briefly recalled in Section 2, causes considerable technical difficulties that were not present in our previous work [26] and makes the analysis of probabilistic properties of Gaussian polytopes a demanding task.…”

“…In particular, for any fixed λ there is no centred ball with radius only depending on λ (or any other deterministic set that depends on the parameter λ only) in which a Gaussian polytope is included with probability one. This in turn implies that the scaling transformation we borrow from [10], which we recall in Section 2 below, maps a Gaussian polytope into a random set in the product space R d−1 × R, while the scaling transformation for random polytopes in the unit ball has R d−1 × [0, ∞) as its target space, see [9]. Here, the upper half-space R d−1 × [0, ∞) corresponds to the image of an appropriate centred ball that contains the Gaussian polytope with high probability, while the lower half-space R d−1 × (−∞, 0) corresponds to the image of its complement.…”

“…The exponential map exp := exp u 0 : T u 0 → S d−1 maps a vector v ∈ T u 0 to the point u ∈ S d−1 in such a way that u lies at the end of the unique geodesic ray with length v starting at u 0 and having direction v. We denote by B d−1 (0, π) := {v ∈ T u 0 : v < π} the region of T u 0 on which the exponential map is injective and note that exp(B d−1 (0, π)) = S d−1 \ {−u 0 }. (Following [10], we prefer to write B d−1 (0, r) for a centred ball of radius r > 0 in T u 0 instead of B d−1 (0, r) to prevent confusions. )…”

“…Because of the mapping property of Poisson point processes, the rescaled point process P (λ) is a Poisson point process on W λ . In particular, Equation (3.18) in [10] shows that the intensity measure of P (λ) has density…”

“…In Section 1.2 we present our main findings for the volume and the face numbers of Gaussian polytopes. Some preliminaries, especially the scaling transformation from [10] as well as the measure-valued versions of our functionals are described in Section 2. We also introduce there a cluster measure representation of their cumulant measures, that will turn out to be crucial for later purposes.…”