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We study divisibility properties of a set { f 1 (U n (s)) , ... , f m (U n (s)) } , where f 1 , ... , f m are polynomials in s variables over Z and U n (s) is a point picked uniformly at random from the set { 1 , ... , n } s. We show that, as n → ∞ , the GCD and the suitably normalized LCM of this set converge in distribution to a.s. finite random variables under mild assumptions on f 1 , ... , f m. Our approach is based on the known fact that the uniform distribution on { 1 , ... , n } converges to the Haar measure on the ring Z ˆ of profinite integers, combined with the Lang–Weil bounds and tools from probability theory. [ABSTRACT FROM AUTHOR]

We introduce and study the r -Lah distribution whose definition involves r -Stirling numbers of both kinds. We compute its expectation and variance, show its log-concavity and prove limit theorems for this distribution. We use these results to prove threshold phenomena for convex cones generated by random walks and to analyze the probability of unique recovery of sparse monotone signals from linear measurements. [ABSTRACT FROM AUTHOR]

Consider a sequence of partial sums S i = ξ 1 + … + ξ i , 1 ≤ i ≤ n , starting at S 0 = 0 , whose increments ξ 1 , … , ξ n are random vectors in R d , d ≤ n . We are interested in the properties of the convex hull C n : = Conv ( S 0 , S 1 , … , S n ) . Assuming that the tuple ( ξ 1 , … , ξ n ) is exchangeable and a certain general position condition holds, we prove that the expected number of k -dimensional faces of C n is given by the formula E [ f k ( C n ) ] = 2 ⋅ k ! n ! ∑ l = 0 ∞ [ n + 1 d − 2 l ] { d − 2 l k + 1 } , for all 0 ≤ k ≤ d − 1 , where [ n m ] and { n m } are Stirling numbers of the first and second kind, respectively. Further, we compute explicitly the probability that for given indices 0 ≤ i 1 < … < i k + 1 ≤ n , the points S i 1 , … , S i k + 1 form a k -dimensional face of Conv ( S 0 , S 1 , … , S n ) . This is done in two different settings: for random walks with symmetrically exchangeable increments and for random bridges with exchangeable increments. These results generalize the classical one-dimensional discrete arcsine law for the position of the maximum due to E. Sparre Andersen. All our formulae are distribution-free, that is do not depend on the distribution of the increments ξ k 's. The main ingredient in the proof is the computation of the probability that the origin is absorbed by a joint convex hull of several random walks and bridges whose increments are invariant with respect to the action of direct product of finitely many reflection groups of types A n − 1 and B n . This probability, in turn, is related to the number of Weyl chambers of a product-type reflection group that are intersected by a linear subspace in general position. [ABSTRACT FROM AUTHOR]

Motivated by problems of hyperbolic stochastic geometry we introduce and study the class of beta-star polytopes. A beta-star polytope is defined as the convex hull of an inhomogeneous Poisson processes on the complement of the unit ball in R d with density proportional to (‖ x ‖ 2 − 1) − β , where ‖ x ‖ > 1 and β > d / 2. Explicit formulas for various geometric and combinatorial functionals associated with beta-star polytopes are provided, including the expected number of k -dimensional faces, the expected external angle sums and the expected intrinsic volumes. Beta-star polytopes are relevant in the context of hyperbolic stochastic geometry, since they are tightly connected to the typical cell of a Poisson-Voronoi tessellation as well as the zero cell of a Poisson hyperplane tessellation in hyperbolic space. The general results for beta-star polytopes are used to provide explicit formulas for the expected f -vector of the typical hyperbolic Poisson-Voronoi cell and the hyperbolic Poisson zero cell. Their asymptotics for large intensities and their monotonicity behaviour is discussed as well. Finally, stochastic geometry in the de Sitter half-space is studied as the hyperbolic analogue to recent investigations about random cones generated by random points on half-spheres in spherical or conical stochastic geometry. [ABSTRACT FROM AUTHOR]

Pick d + 1 points uniformly at random on the unit sphere in R d. What is the expected value of the angle sum of the simplex spanned by these points? Choose n points uniformly at random in the d -dimensional ball. What is the expected number of faces of their convex hull? We answer these and some other, seemingly unrelated, questions of stochastic geometry. To this end, we compute expected internal angles of random simplices whose vertices are independent random points sampled from one of the following d -dimensional distributions: (i) the beta distribution with the density proportional to (1 − ‖ x ‖ 2) β , where x is belongs to the unit ball in R d ; (ii) the beta' distribution with the density proportional to (1 + ‖ x ‖ 2) − β on the whole of R d. These results imply explicit formulae for the expected face numbers of the following random polytopes: (a) the typical Poisson-Voronoi cell; (b) the zero cell of the Poisson hyperplane tessellation; (c) beta and beta' polytopes defined as convex hulls of i.i.d. samples from the corresponding distributions. [ABSTRACT FROM AUTHOR]

We study the precise asymptotic volume of balls in Orlicz spaces and show that the volume of the intersection of two Orlicz balls undergoes a phase transition when the dimension of the ambient space tends to infinity. This generalizes a result of Schechtman and Schmuckenschläger (1991) [32] for ℓ p d -balls. As another application, we determine the precise asymptotic volume ratio for 2-concave Orlicz spaces ℓ M d. Our method rests on ideas from statistical mechanics and large deviations theory, more precisely the maximum entropy or Gibbs principle for non-interacting particles, and presents a natural approach and fresh perspective to such geometric and volumetric questions. In particular, our approach explains how the p -generalized Gaussian distribution occurs in problems related to the geometry of ℓ p d -balls, which are Orlicz balls when the Orlicz function is M (t) = | t | p. [ABSTRACT FROM AUTHOR]

We study random polytopes of the form [ X 1 , ... , X n ] defined as convex hulls of independent and identically distributed random points X 1 , ... , X n in R d with one of the following densities: f d , β (x) = c d , β (1 − ‖ x ‖ 2) β , ‖ x ‖ < 1 , (beta distribution, β > − 1) or f ˜ d , β (x) = c ˜ d , β (1 + ‖ x ‖ 2) − β , x ∈ R d , (beta' distribution, β > d / 2). Here, c d , β and c ˜ d , β are normalizing constants. This setting also includes the uniform distribution on the unit sphere and the standard normal distribution as limiting cases. We derive exact and asymptotic formulae for the expected number of k -faces of [ X 1 , ... , X n ] for arbitrary k ∈ { 0 , 1 , ... , d − 1 }. We prove that for any such k this expected number is strictly monotonically increasing with n. Also, we compute the expected internal and external angles of these polytopes at faces of every dimension and, more generally, the expected conic intrinsic volumes of their tangent cones. By passing to the large n limit in the beta' case, we compute the expected f -vector of the convex hull of Poisson point processes with power-law intensity function. Using convex duality, we derive exact formulae for the expected number of k -faces of the zero cell for a class of isotropic Poisson hyperplane tessellations in R d. This family includes the zero cell of a classical stationary and isotropic Poisson hyperplane tessellation and the typical cell of a stationary Poisson–Voronoi tessellation as special cases. In addition, we prove precise limit theorems for this f -vector in the high-dimensional regime, as d → ∞. Finally, we relate the d -dimensional beta and beta' distributions to the generalized Pareto distributions known in extreme-value theory. [ABSTRACT FROM AUTHOR]

The unit ball B p n (R) of the finite-dimensional Schatten trace class S p n consists of all real n × n matrices A whose singular values s 1 (A) , ... , s n (A) satisfy s 1 p (A) + ... + s n p (A) ≤ 1 , where p > 0. Saint Raymond (1984) showed that the limit lim n → ∞ n 1 ∕ 2 + 1 ∕ p (Vol B p n (R)) 1 ∕ n 2 exists in (0 , ∞) and provided both lower and upper bounds. In this manuscript we use the theory of logarithmic potentials in external fields to determine the precise limiting constant and thus the exact asymptotic volume of B p n (R). The corresponding result for complex Schatten balls is also obtained. As an application we compute the precise asymptotic volume ratio of the Schatten p -balls, as n → ∞ , thereby extending Saint Raymond's estimate in the case of the nuclear norm (p = 1) to the full regime 1 ≤ p ≤ ∞ with exact limiting behavior. [ABSTRACT FROM AUTHOR]