Related papers: Random Gaussian Tetrahedra
Let us consider the set $\Omega (\triangle ABC)$ of all tetrahedra $ABCD$ with a given non-degenerate base $ABC$ in $\mathbb{E}^3$ and $D$ lying outside the plane $ABC$. Let us denote by $\Sigma(\triangle ABC)$ the set $\left\{\Bigl(\cos…
In this paper we give a formula for the probability that $n$ random points chosen under the uniform distribution in a disk are in convex position. While close, the formula is recursive and is totally explicit only for the first values of…
For a graph G, let p_i(G), i=0,...,3 be the probability that three distinct random vertices span exactly i edges. We call (p_0(G),...,p_3(G)) the 3-local profile of G. We investigate the set ${\cal S}_3 \subset \mathbb R^4$ of all vectors…
Consider two half-spaces $H_1^+$ and $H_2^+$ in $\mathbb{R}^{d+1}$ whose bounding hyperplanes $H_1$ and $H_2$ are orthogonal and pass through the origin. The intersection $\mathbb{S}_{2,+}^d:=\mathbb{S}^d\cap H_1^+\cap H_2^+$ is a spherical…
We consider a random variable expressed as the Euclidean distance between an arbitrary point and a random variable uniformly distributed in a closed and bounded set of a three-dimensional Euclidean space. Four cases are considered for this…
The curse of dimensionality is a common phenomenon which affects analysis of datasets characterized by large numbers of variables associated with each point. Problematic scenarios of this type frequently arise in classification algorithms…
The prediction of the final state probabilities of a general cuboid randomly thrown onto a surface is a problem that naturally arises in the minds of men and women familiar with regular cubic dice and the basic concepts of probability.…
We consider the 2-dimensional random matching problem in $\mathbb{R}^2.$ In a challenging paper, Caracciolo et. al. arXiv:1402.6993 on the basis of a subtle linearization of the Monge Ampere equation, conjectured that the expected value of…
Consider a random walk in $\mathbb{R}^d$ that starts at the origin and whose increment distribution assigns zero probability to any affine hyperplane. We solve Sylvester's problem for these random walks by showing that the probability that…
Settling a first case of a conjecture of M. Kahle on the homology of the clique complex of the random graph $G=G_{n,p}$, we show, roughly speaking, that (with high probability) the triangles of $G$ span its cycle space whenever each of its…
We introduce a model for a growing random graph based on simultaneous reproduction of the vertices. The model can be thought of as a generalisation of the reproducing graphs of Southwell and Cannings and Bonato et al to allow for a random…
We count the asymptotic number of triangles in uniform random graphs where the degree distribution follows a power law with degree exponent $\tau\in(2,3)$. We also analyze the local clustering coefficient $c(k)$, the probability that two…
In this report, the explicit probability density functions of the random Euclidean distances associated with regular hexagons are given, when the two endpoints of a link are randomly distributed in the same hexagon, and two adjacent…
We prove a pointwise version of the multi-dimensional central limit theorem for convex bodies. Namely, let X be an isotropic random vector in R^n with a log-concave density. For a typical subspace E in R^n of dimension n^c, consider the…
Given a set C in R^d, let p(C) be the probability that a random d-dimensional unimodular lattice, chosen according to Haar measure on SL(d,Z)\SL(d,R), is disjoint from C\{0}. For special convex sets C we prove bounds on p(C) which are sharp…
We compute the path integral of three-dimensional gravity with negative cosmological constant on spaces which are topologically a torus times an interval. These are Euclidean wormholes, which smoothly interpolate between two asymptotically…
We consider the set of points chosen randomly, independently and uniformly in the $d$-dimensional spherical layer. A set of points is called $1$-convex if all its points are vertices of the convex hull of this set. In \cite{3} an estimate…
The algebraic area probability distribution of closed planar random walks of length N on a square lattice is considered. The generating function for the distribution satisfies a recurrence relation in which the combinatorics is encoded. A…
What is the probability that a random UHF algebra is of infinite type? What is the probability that a random simple AI algebra has at most $k$ extremal traces? What is the expected value of the radius of comparison of a random…
We present new examples of topologically convex edge-ununfoldable polyhedra, i.e., polyhedra that are combinatorially equivalent to convex polyhedra, yet cannot be cut along their edges and unfolded into one planar piece without overlap.…