相关论文: Random Convex Hulls and Extreme Value Statistics

We establish an exact formula for the average number of edges appearing on the boundary of the global convex hull of n independent Brownian paths in the plane. This requires the introduction of a counting criterion which amounts to "cutting…

For the perimeter length $L_n$ and the area $A_n$ of the convex hull of the first $n$ steps of a planar random walk, this thesis study $n \to \infty$ mean and variance asymptotics and establish distributional limits. The results apply to…

The convex hull of a set of points, $C$, serves to expose extremal properties of $C$ and can help identify elements in $C$ of high interest. For many problems, particularly in the presence of noise, the true vertex set (and facets) may be…

We study the convex hulls of reachable sets of nonlinear systems with bounded disturbances and uncertain initial conditions. Reachable sets play a critical role in control, but remain notoriously challenging to compute, and existing…

Convex records have an appealing purely geometric definition. In a sequence of $d$-dimensional data points, the $n$-th point is a convex record if it lies outside the convex hull of all preceding points. We specifically focus on the…

For a $d$-dimensional random vector $X$, let $p_{n, X}(\theta)$ be the probability that the convex hull of $n$ independent copies of $X$ contains a given point $\theta$. We provide several sharp inequalities regarding $p_{n, X}(\theta)$ and…

In this thesis, we study three physically relevant models of strongly correlated random variables: trapped fermions, random matrices and random walks. In the first part, we show several exact mappings between the ground state of a trapped…

In this paper we present several results on the expected complexity of a convex hull of $n$ points chosen uniformly and independently from a convex shape. (i) We show that the expected number of vertices of the convex hull of $n$ points,…

We investigate the utility of the convex hull of many Lagrangian tracers to analyze transport properties of turbulent flows with different anisotropy. In direct numerical simulations of statistically homogeneous and stationary Navier-Stokes…

In this work we deal with the problem of support estimation under shape restrictions. The shape restriction we deal with is an extension of the notion of convexity named alpha-convexity. Instead of assuming, as in the convex case, the…

Let $S_k$ be a random walk in $R^d$ such that its distribution of increments does not assign mass to hyperplanes. We study the probability $p_n$ that the convex hull $conv (S_1, \ldots , S_n)$ of the first $n$ steps of the walk does not…

In this paper, we study the limiting behavior of the perimeter and diameter functionals of the convex hull spanned by the first $n$ steps of two planar random walks. As the main results, we obtain the strong law of large numbers and the…

We study various properties of the convex hull of a planar Brownian motion, defined as the minimum convex polygon enclosing the trajectory, in the presence of an infinite reflecting wall. Recently, in a Rapid Communication [Phys. Rev. E…

We analyze the correctness of an O(n log n) time divide-and-conquer algorithm for the convex hull problem when each input point is a location determined by a normal distribution. We show that the algorithm finds the convex hull of such…

A simple symmetric random walk in the space $\mathbb{Z}^2$ is considered. The asymptotic behavior as the number of jumps tends to infinity of the probability that a fixed edge of the random walk lies in the polygon that forms the boundary…

The classical theorem of Wendel provides an exact formula for the probability that the convex hull of independent symmetrically distributed vectors in ${\mathbb R}^d$ contains the origin as long as the distributions of the vectors are…

Consider a sequence of partial sums $S_i= \xi_1+\dots+\xi_i$, $1\leq i\leq n$, starting at $S_0=0$, whose increments $\xi_1,\dots,\xi_n$ are random vectors in $\mathbb R^d$, $d\leq n$. We are interested in the properties of the convex hull…

We investigate reflected random walks in the quarter plane, with particular emphasis on the time spent along the reflection boundary axes. Assuming the drift of the random walk lies within the cone, the local time converges -- without the…

We examine how the measure and the number of vertices of the convex hull of a random sample of $n$ points from an arbitrary probability measure in $\mathbf{R}^d$ relates to the wet part of that measure. This extends classical results for…

We give an explicit formula for the probability that the convex hull of an $n$-step random walk in $R^d$ does not contain the origin, under the assumption that the distribution of increments of the walk is centrally symmetric and puts no…