For a determinantal point process (DPP) X with a kernel K whose spectrum is strictly less than one, André Goldman has established a coupling to its reduced Palm process $X^u$ at a point u with $K(u,u)>0$ so that, almost surely, $X^u$ is obtained by removing a finite number of points from X. We sharpen this result, assuming weaker conditions and establishing that $X^u$ can be obtained by removing at most one point from X, where we specify the distribution of the difference $\xi_u: = X\setminus X^u$. This is used to discuss the degree of repulsiveness in DPPs in terms of $\xi_u$, including Ginibre point processes and other specific parametric models for DPPs.
For a determinantal point process X with a kernel K whose spectrum is strictly less than one, André Goldman has established a coupling to its reduced Palm process X u at a point u with K(u, u) > 0 so that almost surely X u is obtained by removing a finite number of points from X. We sharpen this result, assuming weaker conditions and establishing that X u can be obtained by removing at most one point from X, where we specify the distribution of the difference ξu := X \ X u . This is used for discussing the degree of repulsiveness in DPPs in terms of ξu, including Ginibre point processes and other specific parametric models for DPPs. Keywords: Ginibre point process; globally most repulsive determinantal point process; isotropic determinantal point process on the sphere; globally most repulsive determinantal point process; projection kernel; stationary determinantal point process in Euclidean space.
Facets of the convex hull of n independent random vectors chosen uniformly at random from the unit sphere in R d are studied. A particular focus is given on the height of the facets as well as the expected number of facets as the dimension increases. Regimes for n and d with different asymptotic behavior of these quantities are identified and asymptotic formulas in each case are established. Extensions of some known results in fixed dimension to the case where dimension tends to infinity are described.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.