Related papers: Almost sharp sharpness for Poisson Boolean percola…
This note shows how to considerably strengthen the usual mode of convergence of an $n$-particle system to its McKean-Vlasov limit, often known as propagation of chaos, when the volatility coefficient is nondegenerate and involves no…
We establish that the phase transition for infinite cycles in the random stirring model on an infinite regular tree of high degree is sharp. That is, we prove that there exists d_0 such that, for any d \geq d_0, the set of parameter values…
Discrete delta functions define the limits of attainable spatial resolution for all imaging systems. Here we construct broad, multi-dimensional discrete functions that replicate closely the action of a Dirac delta function under aperiodic…
We first study a $d$-dimensional branching Brownian motion (BBM) among mild Poissonian obstacles, where a random trap field in $\mathbb{R}^d$ is created via a Poisson point process. The trap field consists of balls of fixed radius centered…
We prove that for Voronoi percolation on $\mathbb{R}^d$, there exists $p_c\in[0,1]$ such that - for $p<p_c$, there exists $c_p>0$ such that $\mathbb{P}_p[0\text{ connected to distance }n]\leq \exp(-c_p n)$, - there exists $c>0$ such that…
The two-parameter Poisson--Dirichlet distribution is a probability distribution on the totality of positive decreasing sequences with sum 1 and hence considered to govern masses of a random discrete distribution. A characterization of the…
Motivated by its relevance for the study of perturbations of one-dimensional voter models, including stochastic Potts models at low temperature, we consider diffusively rescaled coalescing random walks with branching and killing. Our main…
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…
We study the number of real roots of a Kostlan random polynomial of degree $d$ in one variable. More generally, we are interested in the distribution of the counting measure of the set of real roots of such a polynomial. We compute the…
This thesis considers three models which describe a multihop ad-hoc telecommunication system. These systems consist of users sending messages, which can jump to other users to reach the target user. The first two models have already been…
The two parameter Poisson-Dirichlet distribution $PD(\alpha,\theta)$ is the distribution of an infinite dimensional random discrete probability. It is a generalization of Kingman's Poisson-Dirichlet distribution. The two parameter Dirichlet…
Bootstrap percolation on an arbitrary graph has a random initial configuration, where each vertex is occupied with probability p, independently of each other, and a deterministic spreading rule with a fixed parameter k: if a vacant site has…
We investigate spatial random graphs defined on the points of a Poisson process in $d$-dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point is assigned an independent weight. Given the…
One aspect of Poisson approximation is that the support of the random variable of interest is often finite while the support of the Poisson distribution is not. In this paper we will remedy this by examining truncated negative binomial (of…
We consider the random walk loop soup on the discrete half-plane and study the percolation problem, i.e. the existence of an infinite cluster of loops. We show that the critical value of the intensity is equal to 1/2. The absence of…
We study the problem of estimating the parameters of a Boolean product distribution in $d$ dimensions, when the samples are truncated by a set $S \subset \{0, 1\}^d$ accessible through a membership oracle. This is the first time that the…
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…
We study the percolation time of the $r$-neighbour bootstrap percolation model on the discrete torus $(\Z/n\Z)^d$. For $t$ at most a polylog function of $n$ and initial infection probabilities within certain ranges depending on $t$, we…
We consider a branching Brownian motion in $\mathbb{R}^d$ with $d \geq 1$ in which the position $X_t^{(u)}\in \mathbb{R}^d$ of a particle $u$ at time $t$ can be encoded by its direction $\theta^{(u)}_t \in \mathbb{S}^{d-1}$ and its distance…
In r-neighbour bootstrap percolation on a graph G, a set of initially infected vertices A \subset V(G) is chosen independently at random, with density p, and new vertices are subsequently infected if they have at least r infected…