Related papers: Existence of Gibbsian point processes with geometr…
In this paper, we prove the existence of infinite Gibbs Delaunay Potts tessellations for marked particle configurations. The particle systems has two types of interaction, a so-called \emph{background potential} ensures that small and large…
We consider an infinite system of Brownian motions which interact through a given Brownian motion being reflected from its left neighbor. Earlier we studied this system for deterministic periodic initial configurations. In this contribution…
Making use of recent techniques in the theory of selfadjoint extensions of symmetric operators, we characterize the class of point interaction Hamiltonians in a 3-D bounded domain with regular boundary. In the particular case of one point…
We establish phase transitions for classes of continuum Delaunay multi-type particle systems (continuum Potts models) with infinite range repulsive interaction between particles of different type. In one class of the Delaunay Potts models…
Special case of a Gibbsian facet process on a fixed window with a discrete orientation distribution and with increasing intensity of the underlying Poisson process is studied. All asymptotic moments for interaction U-statistics are…
We study the structure of stationary non equilibrium states for interacting particle systems from a microscopic viewpoint. In particular we discuss two different discrete geometric constructions. We apply both of them to determine non…
This work lies at the intersection of Gibbs models and hyperuniform point processes. Classical Gibbs models, whether defined on lattices or in continuous space, provide flexible tools to describe interacting particle systems but are…
We propose a method for detecting significant interactions in very large multivariate spatial point patterns. This methodology develops high dimensional data understanding in the point process setting. The method is based on modelling the…
A new direct integration method is established to construct the solutions of the stationary BBGKY hierarchy, assuming the usual form of the equilibrium correlation functions, for infinite classical systems of particles interacting via a…
Let $P$ be a collection of $n$ points moving along pseudo-algebraic trajectories in the plane. One of the hardest open problems in combinatorial and computational geometry is to obtain a nearly quadratic upper bound, or at least a subcubic…
Asymptotic behavior of the point process of high and medium values of a Gaussian stationary process with discrete time is considered. An approximation by a Poisson cluster point process is given for the point process.
Gravitational and electromagnetic interactions are Hamiltonian systems with forces between pairs of particles. We propose an alternative: Hamiltonian dynamics with triplet interactions between point particles. Our system has a potential…
Networks are a widely used and efficient paradigm to model real-world systems where basic units interact pairwise. Many body interactions are often at play, and cannot be modelled by resorting to binary exchanges. In this work, we consider…
We describe all countable particle systems on $\mathbb{R}$ which have the following three properties: independence, Gaussianity and stationarity. More precisely, we consider particles on the real line starting at the points of a Poisson…
In recent years, there has been considerable interest in developing machine learning models on graphs to account for topological inductive biases. In particular, recent attention has been given to Gaussian processes on such structures since…
We present a new and simple randomized algorithm for constructing the Delaunay triangulation using nearest neighbor graphs for point location. Under suitable assumptions, it runs in linear expected time for points in the plane with…
We consider the Voronoi tessellation associated to a stationary simple point process on $\mathbb{R}^d$ with finite and positive intensity. We introduce the Delaunay triangulation as its dual graph, i.e.~the graph with vertex set given by…
We prove that the number of points of a stationary linear Hawkes process lying in any bounded subset of the real line has exponential moments, without any other assumption than the one needed for existence of such stationary process, namely…
We introduce a new class of dynamic point process models with simple and intuitive dynamics that are based on the Voronoi tessellations generated by the processes. Under broad conditions, these processes prove to be ergodic and produce, on…
We study first-order phase transitions in continuum Gibbs point processes with saturated interactions. These interactions form a broad class of Hamiltonians in which the local energy in regions of high particle density depends only on the…