Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userUniversal Scaling Limits for Generalized Gamma Polytopes
69
0
0.0
(
0
)
Authors
Julian Grote
Ask ChatGPT about the research
No Arabic abstract
Fix a space dimension $dge 2$, 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||^alpha, expleft(-frac{|x|^beta}{beta}right), qquad xin 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 $lambdagamma_{d,alpha,beta}$, $lambda>0$. We establish that the scaling limit of the boundary of $K_lambda$, as $lambda rightarrow infty$, 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 $K_lambda$.
rate research
Read More
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 $n to infty$ for the re-scaled intrinsic volumes and $k$-face functionals of $K_n$, $k in {0,1,...,d-1}$, resolving an open problem. Variance asymptotics are given in terms of functionals of germ-grain models having parabolic grains with apices at a Poisson point process on $R^{d-1} times R$ with intensity $e^h dh dv$. The scaling limit of the boundary of $K_n$ as $n to infty$ 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.
Let K be a convex set in R d and let K $lambda$ be the convex hull of a homogeneous Poisson point process P $lambda$ of intensity $lambda$ on K. When K is a simple polytope, we establish scaling limits as $lambda$ $rightarrow$ $infty$ for the boundary of K $lambda$ in a vicinity of a vertex of K and we give variance asymptotics for the volume and k-face functional of K $lambda$, k $in$ {0, 1, ..., d -- 1}, resolving an open question posed in [18]. The scaling limit of the boundary of K $lambda$ and the variance asymptotics are described in terms of a germ-grain model consisting of cone-like grains pinned to the extreme points of a Poisson point process on R d--1 $times$ R having intensity $sqrt$ de dh dhdv.
The convex hull generated by the restriction to the unit ball of a stationary Poisson point process in the $d$-dimensional Euclidean space is considered. By establishing sharp bounds on cumulants, exponential estimates for large deviation probabilities are derived and the relative error in the central limit theorem on a logarithmic scale is investigated for a large class of key geometric characteristics. This includes the number of lower-dimensional faces and the intrinsic volumes of the random polytopes. Furthermore, moderate deviation principles for the spatial empirical measures induced by these functionals are also established using the method of cumulants. The results are applied to deduce, by duality, fine probabilistic estimates and moderate deviation principles for combinatorial parameters of a class of zero cells associated with Poisson hyperplane mosaics. As a special case this comprises the typical Poisson-Voronoi cell conditioned on having large inradius.
We consider the eigenvalues of symplectic elliptic Ginibre matrices which are known to form a Pfaffian point process whose correlation kernel can be expressed in terms of the skew-orthogonal Hermite polynomials. We derive the scaling limits and the convergence rates of the correlation functions at the real bulk/edge of the spectrum, which in particular establishes the local universality at strong non-Hermiticity. Furthermore, we obtain the subleading corrections of the edge correlation kernels, which depend on the non-Hermiticity parameter contrary to the universal leading term. Our proofs are based on the asymptotic behaviour of the complex elliptic Ginibre ensemble due to Lee and Riser as well as on a version of the Christoffel-Darboux identity, a differential equation satisfied by the skew-orthogonal polynomial kernel.
Suppose we choose $N$ points uniformly randomly from a convex body in $d$ dimensions. How large must $N$ be, asymptotically with respect to $d$, so that the convex hull of the points is nearly as large as the convex body itself? It was shown by Dyer-Furedi-McDiarmid that exponentially many samples suffice when the convex body is the hypercube, and by Pivovarov that the Euclidean ball demands roughly $d^{d/2}$ samples. We show that when the convex body is the simplex, exponentially many samples suffice; this then implies the same result for any convex simplicial polytope with at most exponentially many faces.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
