Related papers: The Triangle Condition for the Marked Random Conne…
We study a stochastic spatial epidemic model where the $N$ individuals carry two features: a position and an infection state, interact and move in $\R^d$. In this Markovian model, the evolution of the infection states are described with the…
We introduce and study a family of 2D percolation systems which are based on the bond percolation model of the triangular lattice. The system under study has local correlations, however, bonds separated by a few lattice spacings act…
We analyse the jigsaw percolation process, which may be seen as a measure of whether two graphs on the same vertex set are `jointly connected'. Bollob\'as, Riordan, Slivken and Smith proved that when the two graphs are independent binomial…
The contact process is a particular case of birth-and-death processes on infinite particle configurations. We consider the contact models on locally compact separable metric spaces. We prove the existence of a one-parameter set of invariant…
We study level-set percolation for Gaussian free fields on metric graphs. In two dimensions, we give an upper bound on the chemical distance between the two boundaries of a macroscopic annulus. Our bound holds with high probability…
The state space of our model is the Euclidean space in dimension d = 2. Simultaneously, from all points of a homogeneous Poisson point process, we let grow independent and identically distributed random continuum paths. Each path stops…
Proofs of sharp phase transition and noise sensitivity in percolation have been significantly simplified by the use of randomized algorithms, via the OSSS inequality (proved by O'Donnell, Saks, Schramm and Servedio (2005)) and the…
We consider self-loops and multiple edges in the configuration model as the size of the graph tends to infinity. The interest in these random variables is due to the fact that the configuration model, conditioned on being simple, is a…
We are interested in phase transitions in certain percolation models on point processes and their dependence on clustering properties of the point processes. We show that point processes with smaller void probabilities and factorial moment…
We introduce a process where a connected rooted multigraph evolves by splitting events on its vertices, occurring randomly in continuous time. When a vertex splits, its incoming edges are randomly assigned between its offspring and a…
In this paper, we introduce a novel model for random hypergraphs based on weighted random connection models. In accordance with the standard theory for hypergraphs, this model is constructed from a bipartite graph. In our stochastic model,…
We propose a statistical model defined on the three-dimensional diamond network where the splitting of randomly selected nodes leads to a spatially disordered network, with decreasing degree of connectivity. The terminal state, that is…
This paper analyzes statistical properties of the Poisson line Cox point process useful in the modeling of vehicular networks. The point process is created by a two-stage construction: a Poisson line process to model road infrastructure and…
In this paper, we study invariant Poisson processes of lines (i.e, bi-infinite geodesics) in the $3$-regular tree. More precisely, there exists a unique (up to multiplicative constant) locally finite Borel measure on the space of lines that…
Consider a discrete locally finite subset $\Gamma$ of $R^d$ and the complete graph $(\Gamma,E)$, with vertices $\Gamma$ and edges $E$. We consider Gibbs measures on the set of sub-graphs with vertices $\Gamma$ and edges $E'\subset E$. The…
The analysis of the linearization effect in multifractal analysis, and hence of the estimation of moments for multifractal processes, is revisited borrowing concepts from the statistical physics of disordered systems, notably from the…
The Vlasov-Poisson system for ions is a kinetic equation for dilute, unmagnetised plasma. It describes the evolution of the ions in a plasma under the assumption that the electrons are thermalized. Consequently, the Poisson coupling for the…
A first-order percolation transition, called explosive percolation, was recently discovered in evolution networks with random edge selection under a certain restriction. However, the network percolation with more realistic evolution…
We study percolation in the following random environment: let $Z$ be a Poisson process of constant intensity in the plane, and form the Voronoi tessellation of the plane with respect to $Z$. Colour each Voronoi cell black with probability…
Based on numerical simulation and local stability analysis we describe the structure of the phase space of the edge/triangle model of random graphs. We support simulation evidence with mathematical proof of continuity and discontinuity for…