Related papers: Size dependence of the largest distance between ra…
We consider a branching Brownian motion evolving in $\mathbb{R}^d$. We prove that the asymptotic behaviour of the maximal displacement is given by a first ballistic order, plus a logarithmic correction that increases with the dimension $d$.…
We obtain a large deviations principle for the self-intersection local times for a symmetric random walk in dimension d>4. As an application, we obtain moderate deviations for random walk in random sceneries in some region of parameters.
The densest amorphous packing of rigid particles is known as random close packing. It has long been appreciated that higher densities are achieved by using collections of particles with a variety of sizes. The variety of sizes is often…
We study the asymptotic behaviour of the maximum interpoint distance of random points in a planar bounded set with an unique major axis and a boundary behaving like an ellipse at the endpoints. Our main result covers the case of uniformly…
Let \( D \) be a strongly connected digraph. The average distance of a vertex \( v \) in \( D \) is defined as the arithmetic mean of the distances from \( v \) to all other vertices in \( D \). The remoteness \( \rho(D) \) of \( D \) is…
Finding the most powerful node in a dynamic random network, the largest set in a partition-valued stochastic process, or the largest family in an evolving population at a given time, can be a very difficult problem. This is particularly the…
For a sample of absolutely bounded i.i.d. random variables with a continuous density the cumulative distribution function of the sample variance is represented by a univariate integral over a Fourier series. If the density is a polynomial…
Nearest-neighbor methods are fundamental to classical and modern machine learning, yet their geometric properties are typically analyzed under independent sampling. In this paper, we study the nearest-neighbor radii under dependent…
Let $X_1,X_2, \ldots $ be independent random uniform points in a bounded domain $A \subset \mathbb{R}^d$ with smooth boundary. Define the coverage threshold $R_n$ to be the smallest $r$ such that $A$ is covered by the balls of radius $r$…
We consider a large family of branching-selection particle systems. The branching rate of each particle depends on its rank and is given by a function $b$ defined on the unit interval. There is also a killing measure $D$ supported on the…
We revisit the maximum range sum (MaxRS) problem: given a set $P$ of $n$ weighted points in $\mathbb{R}^d$ and a range $Q$ (typically axis-aligned $d$-box or $d$-ball), the goal is to place $Q$ to maximize the total weight of points in…
Systems which consist of many localized constituents interacting with each other can be represented by complex networks. Consistently, network science has become highly popular in vast fields focusing on natural, artificial and social…
We give an upper bound in O(d ^((n+1)/2)) for the number of critical points of a normal random polynomial with degree d and at most n variables. Using the large deviation principle for the spectral value of large random matrices we obtain…
We study the dual variants of the Erd\H{o}s's distinct distances and unit distance problems. Instead of considering distances determined by points, we consider simplex volumes determined by hyperplanes. We investigate: (1) the maximum…
With the aim of generalizing histogram statistics to higher dimensional cases, density estimation via discrepancy based sequential partition (DSP) has been proposed to learn an adaptive piecewise constant approximation defined on a binary…
We introduce the concept of a hyperuniformity disorder length that controls the variance of volume fraction fluctuations for randomly placed windows of fixed size. In particular, fluctuations are determined by the average number of…
We introduce a simple geometric model which describes the kinetics of fragmentation of d-dimensional objects. In one dimension our model coincides with the random scission model and show a simple scaling behavior in the long-time limit. For…
One of the main contributions of this paper is to illustrate how large deviation theory can be used to determine the equilibrium distribution of a basic droplet model that underlies a number of important models in material science and…
Random dynamical systems (RDS) evolve by a dynamical rule chosen independently with a certain probability, from a given set of deterministic rules. These dynamical systems in an interval reach a steady state with a unique well-defined…
A set of points with finite density is constructed in $\mathbb{R}^d$, with $d\geq2$, by adding points to a Poisson process such that any line segment of length $O\left(\varepsilon^{-(d-1)}\ln\varepsilon^{-1}\right)$ in $\mathbb{R}^d$ will…