Related papers: Freezing and decorated Poisson point processes
Randomly scaled scale-decorated Poisson point process is introduced recently in Bhattacharya et al. [2017] where it appeared as weak limit of a sequence of point processes in the context of branching random walk. In this article, we obtain…
We consider a two-speed branching random walk, which consists of two macroscopic stages with different reproduction laws. We prove that the centered maximum converges in law to a Gumbel variable with a random shift and the extremal process…
In this paper, we construct scaling limits of some branching random walks in random environment whose off-spring distributions have infinite variance. The Laplace functional of the obtained random measure is given by a non-linear PAM, whose…
We investigate the extremal process of four-dimensional membrane models as the size of the lattice $N$ tends to infinity. We prove the cluster-like geometry of the extreme points and the existence as well as the uniqueness of the extremal…
The extremal process of a branching random walk is the point measure recording the position of particles alive at time $n$, shifted around the expected position of the minimal position. Madaule proved that this point measure converges, as…
In comparison with Derrida's REM, we investigate the influence of the so-called decoration processes arising in the limiting extremal processes of numerous log-correlated Gaussian fields. In particular, we focus on the branching Brownian…
This paper is concerned with the limit laws of the extreme order statistics derived from a symmetric Laplace walk. We provide two different descriptions of the point process of the limiting extreme order statistics: a branching…
We consider a one dimensional random-walk-like process, whose steps are centered Gaussians with variances which are determined according to the sequence of arrivals of a Poisson process on the line. This process is decorated by independent…
We prove that the extremal process of branching Brownian motion, in the limit of large times, converges weakly to a cluster point process. The limiting process is a (randomly shifted) Poisson cluster process, where the positions of the…
It has been conjectured since the work of Lalley and Sellke (1987) that the branching Brownian motion seen from its tip (e.g. from its rightmost particle) converges to an invariant point process. Very recently, it emerged that this can be…
As a first step toward a characterization of the limiting extremal process of branching Brownian motion, we proved in a recent work [Comm. Pure Appl. Math. 64 (2011) 1647-1676] that, in the limit of large time $t$, extremal particles…
We construct the conditional version of $k$ independent and identically distributed random walks on $\R$ given that they stay in strict order at all times. This is a generalisation of so-called non-colliding or non-intersecting random…
We consider the limiting extremal process ${\mathcal X}$ of the particles of the binary branching Brownian motion. We show that after a shift by the logarithm of the derivative martingale $Z$, the rescaled "density" of particles, which are…
In this article, we study the extremal processes of branching Brownian motions conditioned on having an unusually large maximum. The limiting point measures form a one-parameter family and are the decoration point measures in the extremal…
This paper gives an elementary proof for the following theorem: a renewal process can be represented by a doubly-stochastic Poisson process (DSPP) if and only if the Laplace-Stieltjes transform of the inter-arrival times is of the following…
We study a $d$-dimensional branching Brownian motion (BBM) among Poissonian obstacles, where a random trap field in $\mathbb{R}^d$ is created via a Poisson point process. In the soft obstacle model, the trap field consists of a positive…
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…
We consider a continuous time random walk on the two-dimensional discrete torus, whose motion is governed by the discrete Gaussian free field on the corresponding box acting as a potential. More precisely, at any vertex the walk waits an…
We extend the results of Arguin et al and A\"\i{}d\'ekon et al on the convergence of the extremal process of branching Brownian motion by adding an extra dimension that encodes the "location" of the particle in the underlying Galton-Watson…
In this paper, we study branching Brownian motion with absorption, in which particles undergo Brownian motions with drift and are killed upon reaching the origin. We prove that the extremal process of this branching Brownian motion with…