Universal Scaling Limits for Generalized Gamma Polytopes

Abstract

Fix a space dimension $d\ge 2$, parameters $\alpha >-1$ and $\beta \ge 1$, and let ${\gamma }_{d,\alpha ,\beta }$ be the probability measure of an isotropic random vector in ${\mathbb{R}}^d$ with density proportional to \begin{align*} ||x||^α\, \exp\left(-\frac{\|x\|^β}β\right), \qquad x\in \mathbb{R}^d. \end{align*} By $K_{\lambda }$, we denote the Generalized Gamma Polytope arising as the random convex hull of a Poisson point process in ${\mathbb{R}}^d$ with intensity measure $\lambda {\gamma }_{d,\alpha ,\beta }$, $\lambda >0$. We establish that the scaling limit of the boundary of $K_{\lambda }$, as $\lambda \to \infty$, is given by a universal `festoon' of piecewise parabolic surfaces, independent of $\alpha$ and $\beta$. Moreover, we state a list of other large scale asymptotic results, including expectation and variance asymptotics, central limit theorems, concentration inequalities, Marcinkiewicz-Zygmund-type strong laws of large numbers, as well as moderate deviation principles for the intrinsic volumes and face numbers of $K_{\lambda }$.

19 pages, 7 figures