Related papers: Percolative properties of Brownian interlacements …
We consider connectivity properties of the vacant set of (random) ensembles of Wiener sausages in $\mathbb R^d$ in the transient dimensions $d \geq 3$. We prove that the vacant set of Brownian interlacements contains at most one infinite…
We introduce the model of two-dimensional continuous random interlacements, which is constructed using the Brownian trajectories conditioned on not hitting a fixed set (usually, a disk). This model yields the local picture of Wiener sausage…
In this paper we establish some properties of percolation for the vacant set of random interlacements, for d at least 5 and small intensity u. The model of random interlacements was first introduced by A.S. Sznitman in arXiv:0704.2560. It…
The model of random interlacements on Z^d, d bigger or equal to 3, was recently introduced in arXiv:0704.2560. A non-negative parameter u parametrizes the density of random interlacements on Z^d. In the present note we investigate the…
In this article, we first extend the construction of random interlacements, introduced by A.S. Sznitman in [arXiv:0704.2560], to the more general setting of transient weighted graphs. We prove the Harris-FKG inequality for this model and…
Random interlacements at level u is a one parameter family of connected random subsets of Z^d, d>=3 introduced in arXiv:0704.2560. Its complement, the vacant set at level u, exhibits a non-trivial percolation phase transition in u, as shown…
We introduce a model of random interlacements made of a countable collection of doubly infinite paths on Z^d, d bigger or equal to 3. A non-negative parameter u measures how many trajectories enter the picture. This model describes in the…
We prove the existence of non-trivial phase transitions for the intersection of two independent random interlacements and the complement of the intersection. Some asymptotic results about the phase curves are also obtained. Moreover, we…
We investigate random interlacements on Z^d, d bigger or equal to 3. This model recently introduced in arXiv:0704.2560 corresponds to a Poisson cloud on the space of doubly infinite trajectories modulo time-shift tending to infinity at…
The vacant set of random interlacements on ${\mathbb{Z}}^d$, $d\ge3$, has nontrivial percolative properties. It is known from Sznitman [Ann. Math. 171 (2010) 2039--2087], Sidoravicius and Sznitman [Comm. Pure Appl. Math. 62 (2009) 831--858]…
We study the limiting shape of the connected components of the vacant set of two-dimensional Brownian random interlacements: we prove that the connected component around $x$ is close in distribution to a rescaled \emph{Brownian amoeba} in…
The vacant set of random interlacements at level $u>0$, introduced in arXiv:0704.2560, is a percolation model on $\mathbb{Z}^d$, $d \geq 3$ which arises as the set of sites avoided by a Poissonian cloud of doubly infinite trajectories,…
We consider the branching interlacement model introduced by Zhu as an analog of Sznitman's random interlacement for branching random walks. We show that two points of the interlacement are connected via at most $\lceil d/4 \rceil$…
The random interlacements (at level u) is a one parameter family of random subsets of Z^d introduced by Sznitman in arXiv:0704.2560. The vacant set at level u is the complement of the random interlacement at level u. In this paper, we study…
We consider a continuum percolation model on $\R^d$, $d\geq 1$.For $t,\lambda\in (0,\infty)$ and $d\in\{1,2,3\}$, the occupied set is given by the union of independent Brownian paths running up to time $t$ whoseinitial points form a Poisson…
We study the percolative properties of random interlacements on the product of G with the integer line Z, when G is a weighted graph satisfying certain sub-Gaussian estimates attached to the parameters alpha > 1, measuring the volume growth…
The main focus of this article concerns the strongly percolative regime of the vacant set of random interlacements on $ \mathbb{Z}^d$, with $d \ge 3$. We investigate the occurrence in a large box of an excessive fraction of sites that get…
We consider the Brownian interlacements model in Euclidean space, introduced by A.S. Sznitman in \cite{sznitman2013scaling}. We give estimates for the asymptotics of the visibility in the vacant set. We also consider visibility inside the…
We consider continuous-time random interlacements on Z^d, d greater or equal to 3, and investigate the percolation model where a site x of Z^d is occupied if the total amount of time spent at x by all the trajectories of the interlacement…
Partially motivated by the desire to better understand the connectivity phase transition in fractal percolation, we introduce and study a class of continuum fractal percolation models in dimension d greater than or equal to 2. These include…