Related papers: Size dependence of the largest distance between ra…
The sizes and shapes of voids in a galaxy survey depend not only on the physics of structure formation, but also on the sampling density of the survey and on the algorithm used to define voids. Using an N-body simulation with a CDM power…
We consider statistical models arising from the common set of solutions to a sparse polynomial system with general coefficients. The maximum likelihood degree counts the number of critical points of the likelihood function restricted to the…
We consider a random field $\phi(\mathbf{r})$ in $d$ dimensions which is largely concentrated around small `hotspots', with `weights', $w_i$. These weights may have a very broad distribution, such that their mean does not exist, or else is…
We study the large deviation function for the empirical measure of diffusing particles at one fixed position. We find that the large deviation function exhibits anomalous system size dependence in systems that satisfy the following…
We consider the distribution of the sum and the maximum of a collection of independent exponentially distributed random variables. The focus is laid on the explicit form of the density functions (pdf) of non-i.i.d. sequences. Those are…
Let $X_1, \ldots, X_n$ be independent random points drawn from an absolutely continuous probability measure with density $f$ in $\mathbb{R}^d$. Under mild conditions on $f$, we derive a Poisson limit theorem for the number of large…
We study the problem of computing the \textsc{Maxima} of a set of $n$ $d$-dimensional points. For dimensions 2 and 3, there are algorithms to solve the problem with order-oblivious instance-optimal running time. However, in higher…
A random geometric graph $G(\mathcal{X}_n, r_n)$ is formed by taking a binomial process $\mathcal{X}_n$ as the set of vertices and joining any two distinct points with an edge if they lie within distance $r_n$ of each other. We investigate…
We study open point sets in Euclidean spaces $\mathbb{R}^d$ without a pair of points an integral distance apart. By a result of Furstenberg, Katznelson, and Weiss such sets must be of Lebesgue upper density zero. We are interested in how…
We give the distribution of $M_n$, the maximum of a sequence of $n$ observations from a moving average of order 1. Solutions are first given in terms of repeated integrals and then for the case where the underlying independent random…
We study the relativistic Vlasov-Maxwell system which describes large systems of particles interacting by means of their collectively generated forces. If the speed of light $c$ is considered as a parameter then it is known that in the…
Let $G_n$ be a random geometric graph with vertex set $[n]$ based on $n$ i.i.d.\ random vectors $X_1,\ldots,X_n$ drawn from an unknown density $f$ on $\R^d$. An edge $(i,j)$ is present when $\|X_i -X_j\| \le r_n$, for a given threshold…
We consider the question of the largest possible combinatorial diameter among $(d-1)$-dimensional simplicial complexes on $n$ vertices, denoted $H_s(n, d)$. Using a probabilistic construction we give a new lower bound on $H_s(n, d)$ that is…
We study a kinetic mean-field equation for a system of particles with different sizes, in which particles are allowed to coagulate only if their sizes sum up to a prescribed time-dependent value. We prove well-posedness of this model, study…
This paper studies the limits of a spatial random field generated by uniformly scattered random sets, as the density $\lambda$ of the sets grows to infinity and the mean volume $\rho$ of the sets tends to zero. Assuming that the volume…
Consider a set $P$ of $n$ points picked uniformly and independently from $[0,1]^d$ for a constant dimension $d$ -- such a point set is extremely well behaved in many aspects. For example, for a fixed $r \in [0,1]$, we prove a new…
We prove algorithmic and hardness results for the problem of finding the largest set of a fixed diameter in the Euclidean space. In particular, we prove that if $A^*$ is the largest subset of diameter $r$ of $n$ points in the Euclidean…
We present a geometrical model of the distribution of luminous matter in the universe, derived from a very simple reaction-diffusion model of turbulent phenomena. The apparent dimension of luminous matter, $D(l)$, depends linearly on the…
We consider the distribution of free path lengths, or the distance between consecutive bounces of random particles, in an n-dimensional rectangular box. If each particle travels a distance R, then, as R tends to infinity the free path…
The continuous limit of large systems of particles of finite size on the line is described. The particles are assumed to move freely and stick under collision, to form compound particles whose mass and size is the sum of the masses and…