Related papers: Infinite canonical super-Brownian motion and scali…
Scaled Brownian motion (SBM) is widely used to model anomalous diffusion of passive tracers in complex and biological systems. It is a highly non-stationary process governed by the Langevin equation for Brownian motion, however, with a…
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…
We study scaling limits of non-increasing Markov chains with values in the set of non-negative integers, under the assumption that the large jump events are rare and happen at rates that behave like a negative power of the current state. We…
We define the height process for super-critical continuous state branching processes with quadratic branching mechanism. It appears as a projective limit of Brownian motions with positive drift reflected at 0 and a>0 as a goes to infinity.…
The Brownian web is a collection of one-dimensional coalescing Brownian motions starting from every point in space and time, while the Brownian net is an extension that also allows branching. We show here that the Brownian net is the…
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…
The classical definitions of the Incipient Infinite Cluster (IIC) of percolation consist in conditioning the origin on being connected to radius $n$ and letting $n$ go to infinity. We provide a short proof of that convergence in the planar…
We study a limit behavior of a sequence of Markov processes (or Markov chains) such that their distributions outside of any neighborhood of a "singular" point attract to some probability law. In any neighborhood of this point the behavior…
Baxter permutations, plane bipolar orientations, and a specific family of walks in the non-negative quadrant are well-known to be related to each other through several bijections. We introduce a further new family of discrete objects,…
The self-catalytic branching Brownian motions (SBBM) are extensions of the classical one-dimensional branching Brownian motions by incorporating pairwise branchings catalyzed by the intersection local times of the particle pairs. These…
A superprocess with coalescing spatial motion is constructed in terms of one-dimensional excursions. Based on this construction, it is proved that the superprocess is purely atomic and arises as scaling limit of a special form of the…
We consider a super-Brownian motion $X$. Its canonical measures can be studied through the path-valued process called the Brownian snake. We obtain the limiting behavior of the volume of the $\epsilon$-neighborhood for the range of the…
The incipient infinite cluster (IIC) measure is the percolation measure at criticality conditioned on the cluster of the origin to be infinite. Using the lace expansion, we construct the IIC measure for high-dimensional percolation models…
Recently, Holmes and Perkins identified conditions which ensure that for a class of critical lattice models the scaling limit of the range is the range of super-Brownian motion. One of their conditions is an estimate on a spatial moment of…
We consider scaled Brownian motion (sBm), a random process described by a diffusion equation with explicitly time-dependent diffusion coefficient $D(t) = D_0 t^{\alpha - 1}$ (Batchelor's equation) which, for $\alpha < 1$, is often used for…
We consider the simple random walk on the (unique) infinite cluster of super-critical bond percolation in $\Z^d$ with $d\ge2$. We prove that, for almost every percolation configuration, the path distribution of the walk converges weakly to…
Lawler, Schramm and Werner showed that the scaling limit of the loop-erased random walk on $\mathbb{Z}^2$ is $\mathrm{SLE}_2$. We consider scaling limits of the loop-erasure of random walks on other planar graphs (graphs embedded into…
We prove that the total range of Super-Brownian motion with quadratic branching mechanism has an exact packing measure with respect to the gauge function $g(r)=r^4 (\log \log1/r)^{-3}$ in super-critical dimensions $d\geq 5$. More precisely,…
We prove that, after suitable rescaling, the simple random walk on the trace of a large critical branching random walk converges to the Brownian motion on the integrated super-Brownian excursion.
We prove existence of the scaling limit of the invasion percolation cluster (IPC) on a regular tree. The limit is a random real tree with a single end. The contour and height functions of the limit are described as certain diffusive…