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Let \mathcal Z_d be the zero cell of a d-dimensional, isotropic and stationary Poisson hyperplane tessellation. We study the asymptotic behavior of the expected number of k-dimensional faces of \mathcal Z_d, as d\to \infty. For example, we show that the expected number of hyperfaces of \mathcal Z_d is asymptotically equivalent to \sqrt {2\pi /3}\, d^{3/2}, as d\to \infty. We also prove that the expected solid angle of a random cone spanned by d random vectors that are independent and uniformly distributed on the unit upper half-sphere in \mathbb R^{d} is asymptotic to \sqrt 3 \pi ^{-d}, as d\to \infty. [ABSTRACT FROM AUTHOR]

We consider d-dimensional random vectors Y1, . . . ,Yn that satisfy a mild general position assumption a.s. The hyperplanes (Yi−Yj)⊥ (1 ≤ i < j ≤ n) generate a conical tessellation of the Euclidean d-space which is closely related to the Weyl chambers of type An−1. We determine the number of cones in this tessellation and show that it is a.s. constant. For a random cone chosen uniformly at random from this random tessellation, we compute expectations of several geometric functionals. These include the face numbers, as well as the conic intrinsic volumes and the conical quermassintegrals. Under the additional assumption of exchangeability on Y1, . . . ,Yn, the same is done for the dual random cones which have the same distribution as the positive hull of Y1−Y2, . . . , Yn−1−Yn conditioned on the event that this positive hull is not equal to Rd. All these expectations turn out to be distribution-free. Similarly, we consider the conical tessellation induced by the hyperplanes (Yi + Yj)⊥ (1 ≤ i < j ≤ n), (Yi − Yj)⊥ (1 ≤ i < j ≤ n), Yi⊥ (1 ≤ i ≤ n). This tessellation is closely related to the Weyl chambers of type Bn. We compute the number of cones in this tessellation and the expectations of various geometric functionals for random cones drawn from this random tessellation. The main ingredient in the proofs is a connection between the number of faces of the tessellation and the number of faces of the Weyl chambers of the corresponding type that are intersected non-trivially by a certain linear subspace in general position. [ABSTRACT FROM AUTHOR]

Consider a random d-dimensional simplex whose vertices are d + 1 random points sampled independently and uniformly from the unit sphere in Rd. We show that the expected sum of solid angles at the vertices of this random simplex equals 1/8 if d = 3 and 539/288π2 − 1/6 if d = 4. The angles are measured as proportions of the full solid angle which is normalized to be 1. Similar formulae are obtained if the vertices of the simplex are uniformly distributed in the unit ball. These results are special cases of general formulae for the expected angle sums of random beta simplices in dimensions 3 and 4. [ABSTRACT FROM AUTHOR]

Let X1, . . ., Xn be a standard normal sample in Rd. We compute exactly the expected volume of the Gaussian polytope conv [X1, . . ., Xn], the symmetric Gaussian polytope conv [±X1, . . .,±Xn], and the Gaussian zonotope [0,X1] + · · · + [0,Xn] by exploiting their connection to the regular simplex, the regular cross-polytope, and the cube with the aid of Tsirelson’s formula. The expected volumes of these random polytopes are given by essentially the same expressions as the intrinsic volumes and external angles of the regular polytopes. For all of these quantities, we obtain asymptotic formulae which are more precise than the results which were known before. More generally, we determine the expected volumes of some heteroscedastic random polytopes including conv [l1X1, . . ., lnXn] and conv[±l1X1, . . .,±lnXn], where l1, . . ., ln ≥ 0 are parameters, and the intrinsic volumes of the corresponding deterministic polytopes. Finally, we relate the kth intrinsic volume of the regular simplex Δn−1 to the expected maximum of independent standard Gaussian random variables ξ1, . . ., ξn given that the maximum has multiplicity k. Namely, we show that Vk(Δn−1) = (2π)k/2/k! · lim ε↓0 ε1−kE[max{ξ1, . . ., ξn}1{ξ(n) −ξ(n−k+1) ≤ε}], where ξ(1) ≤ ·· · ≤ ξ(n) denote the order statistics. A similar result holds for the cross-polytope if we replace ξ1, . . ., ξn with their absolute values. [ABSTRACT FROM AUTHOR]

Let Pn be an n-dimensional regular polytope from one of the three infinite series (regular simplices, regular crosspolytopes, and cubes). Project Pn onto a random, uniformly distributed linear subspace of dimension d = 2. We prove that the expected number of k-dimensional faces of the resulting random polytope is an increasing function of n. As a corollary, we show that the expected number of k-faces of the Gaussian polytope is an increasing function of the number of points used to generate the polytope. Similar results are obtained for the symmetric Gaussian polytope and the Gaussian zonotope. [ABSTRACT FROM AUTHOR]

A formula due to Sudakov relates the first intrinsic volume of a convex set in a Hilbert space to the maximum of the isonormal Gaussian process over this set. Using this formula we compute the first intrinsic volume of some infinite-dimensional convex compact sets including unit balls with respect to Sobolev-type seminorms and ellipsoids in the Hilbert space. We relate the distribution of the random one-dimensional projections of these sets to the distributions S1, S2, C1, C2 studied by Biane, Pitman, Yor [Bull. AMS 38 (2001)]. We show that the k-th intrinsic volume of the set of all functions on [0, 1] which have Lipschitz constant bounded by 1 and which vanish at 0 (respectively, which have vanishing integral) is given by ... This is related to the results of Gao and Vitale [Discrete Comput. Geom. 26 (2001); Elect. Comm. Probab. 8 (2003)], who considered a similar question for functions with a restriction on the total variation instead of the Lipschitz constant. Using the results of Gao and Vitale we give a new proof of the formula for the expected volume of the convex hull of the d-dimensional Brownian motion which is due to Eldan [Elect. J. Probab. 19 (2014)]. Additionally, we prove an analogue of Eldan's result for the Brownian bridge. Similarly, we show that the results on the intrinsic volumes of the Lipschitz balls can be translated into formulae for the expected volumes of zonoids (Aumann integrals) generated by the Brownian motion and the Brownian bridge. Also, these results have discrete versions for Gaussian random walks and bridges. Our proofs exploit Sudakov's and Tsirelson's theorems which establish a connection between the intrinsic volumes and the isonormal Gaussian process. [ABSTRACT FROM AUTHOR]

Let X1, X2,... be independent identically distributed random variables with values in C. Denote by µ the probability distribution of X1. Consider a random polynomial Pn(z) = (z -- X1)... (z -- Xn). We prove a conjecture of Pemantle and Rivin [in: I. Kotsireas and E. V. Zima, eds., Advances in Combinatorics, Waterloo Workshop in Computer Algebra, 2011] that the empirical measure ... counting the complex zeros of the derivative P'n converges in probability to µ, as n → ∞. [ABSTRACT FROM AUTHOR]

Let (a_k)_{k\in \mathbb N} be an increasing sequence of positive integers satisfying the Hadamard gap condition a_{k+1}/a_k> q >1 for all k\in \mathbb N, and let \begin{equation*} S_n(\omega) = \sum _{k=1}^n \cos (2\pi a_k \omega), \qquad n\in \mathbb N, \; \omega \in [0,1]. \end{equation*} Then S_n is called a lacunary trigonometric sum, and can be viewed as a random variable defined on the probability space \Omega = [0,1] endowed with Lebesgue measure. Lacunary sums are known to exhibit several properties that are typical for sums of independent random variables. For example, a central limit theorem for (S_n)_{n\in \mathbb {N}} has been obtained by Salem and Zygmund, while a law of the iterated logarithm is due to Erdős and Gál. In this paper we study large deviation principles for lacunary sums. Specifically, under the large gap condition a_{k+1}/a_k \to \infty, we prove that the sequence (S_n/n)_{n \in \mathbb {N}} does indeed satisfy a large deviation principle with speed n and the same rate function \widetilde {I} as for sums of independent random variables with the arcsine distribution. On the other hand, we show that the large deviation principle may fail to hold when we only assume the Hadamard gap condition. However, we show that in the special case when a_k= q^k for some q\in \{2,3,\ldots \}, (S_n/n)_{n \in \mathbb {N}} satisfies a large deviation principle (with speed n) and a rate function I_q that is different from \widetilde {I}, and describe an algorithm to compute an arbitrary number of terms in the Taylor expansion of I_q. In addition, we also prove that I_q converges pointwise to \widetilde I as q\to \infty. Furthermore, we construct a random perturbation (a_k)_{k \in \mathbb {N}} of the sequence (2^k)_{k \in \mathbb {N}} for which a_{k+1}/a_k \to 2 as k\to \infty, but for which at the same time (S_n/n)_{n \in \mathbb {N}} satisfies a large deviation principle with the same rate function \widetilde {I} as in the independent case, which is surprisingly different from the rate function I_2 one might naïvely expect. We relate this fact to the number of solutions of certain Diophantine equations. Together, these results show that large deviation principles for lacunary trigonometric sums are very sensitive to the arithmetic properties of the sequence (a_k)_{k\in \mathbb N}. This is particularly noteworthy since no such arithmetic effects are visible in the central limit theorem or in the law of the iterated logarithm for lacunary trigonometric sums. Our proofs use a combination of tools from probability theory, harmonic analysis, and dynamical systems. [ABSTRACT FROM AUTHOR]