Hauke Seidel | Searcharxiv

arXiv2020

Convex cones spanned by regular polytopes

We study three families of polyhedral cones whose sections are regular simplices, cubes, and crosspolytopes. We compute solid angles and conic intrinsic volumes of these cones. We show that several quantities appearing in stochastic geometry can be expressed through these conic intrinsic volumes. A list of such quantities includes internal and external solid angles of regular simplices and crosspolytopes, the probability that a (symmetric) Gaussian random polytope or the Gaussian zonotope contains a given point, the expected number of faces of the intersection of a regular polytope with a random linear subspace passing through its centre, and the expected number of faces of the projection of a regular polytope onto a random linear subspace.

math.PR↗

arXiv2018

Distances between zeroes and critical points for random polynomials with i.i.d. zeroes

Consider a random polynomial $Q_n$ of degree $n+1$ whose zeroes are i.i.d. random variables $ξ_0,ξ_1,\ldots,ξ_n$ in the complex plane. We study the pairing between the zeroes of $Q_n$ and its critical points, i.e. the zeroes of its derivative $Q_n'$. In the asymptotic regime when $n\to\infty$, with high probability there is a critical point of $Q_n$ which is very close to $ξ_0$. We localize the position of this critical point by proving that the difference between $ξ_0$ and the critical point has approximately complex Gaussian distribution with mean $1/(nf(ξ_0))$ and variance of order $\log n \cdot n^{-3}$. Here, $f(z)= \mathbb E[1/(z-ξ_k)]$ is the Cauchy-Stieltjes transform of the $ξ_k$'s. We also state some conjectures on critical points of polynomials with dependent zeroes, for example the Weyl polynomials and characteristic polynomials of random matrices.

math.PR↗