Related papers: Shape and local growth for multidimensional branch…
In this work we prove a shape theorem for a growing set of Simple Random Walks (SRWs), known as frog model. The dynamics of this process is described as follows: There are some active particles, which perform independent SRWs, and sleeping…
We derive a functional central limit theorem for the excursion of a random walk conditioned on sweeping a prescribed geometric area. We assume that the increments of the random walk are integer-valued, centered, with a third moment equal to…
We study the scaling limits of three different aggregation models on the integer lattice Z^d: internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router model, in which particles perform…
Consider a one-dimensional shift-invariant attractive spin-flip system in equilibrium, constituting a dynamic random environment, together with a nearest-neighbor random walk that on occupied sites has a local drift to the right but on…
In this paper, we reveal the branching structure for a non-homogeneous random walk with bounded jumps. The ladder time $T_1,$ the first hitting time of $[1,\infty)$ by the walk starting from $0,$ could be expressed in terms of a…
We study a generalized branching random walk where particles breed at a rate which depends on the number of neighboring particles. Under general assumptions on the breeding rates we prove the existence of a phase where the population…
We first study a model, introduced recently in \cite{ES}, of a critical branching random walk in an IID random environment on the $d$-dimensional integer lattice. The walker performs critical (0-2) branching at a lattice point if and only…
We consider super-diffusive L\'evy walks in $d \geqslant 2$ dimensions when the duration of a single step, i.e., a ballistic motion performed by a walker, is governed by a power-law tailed distribution of infinite variance and finite mean.…
We consider the time evolution of a lattice branching random walk with local perturbations. Under certain conditions, we prove the Carleman type estimation for the moments of a particle subpopulation number and show the existence of a…
Let $S_n$ be a lattice random walk with mean zero and finite variance, and let $\Lambda^a_n$ be its occupation measure at level $a$. In this note, we prove local limit theorems for $\Pr[S_n=x,\Lambda^a_n=\ell]$ and…
A recently developed model of random walks on a $D$-dimensional hyperspherical lattice, where $D$ is {\sl not} restricted to integer values, is used to study polymer growth near a $D$-dimensional attractive hyperspherical boundary. The…
We first establish new local limit estimates for the probability that a nondecreasing integer-valued random walk lies at time $n$ at an arbitrary value, encompassing in particular large deviation regimes. This enables us to derive scaling…
A general multi-type population model is considered, where individuals live and reproduce according to their age and type, but also under the influence of the size and composition of the entire population. We describe the dynamics of the…
The behavior of the maximal displacement of a supercritical branching random walk has been a subject of intense studies for a long time. But only recently the case of time-inhomogeneous branching has gained focus. The contribution of this…
We consider a random walk in a random potential on a square lattice of arbitrary dimension. The potential is a function of an ergodic environment and some steps of the walk. The potential can be unbounded, but it is subject to a moment…
Heterogeneities in environmental conditions often induce corresponding heterogeneities in the distribution of species. In the extreme case of a localized patch of increased growth rates, reproducing populations can become strongly…
We consider a random walk in an i.i.d. random environment on Zd and study properties of its large deviation rate function at the origin. It was proved by Comets, Gantert and Zeitouni in dimension d = 1 in 1999 and later by Varadhan in…
The set of visited sites and the number of visited sites are two basic properties of the random walk trajectory. We consider two independent random walks on a hyper-cubic lattice and study ordering probabilities associated with these…
We consider a certain lattice branching random walk with on-site competition and in an environment which is heterogeneous at a macroscopic scale $1/\varepsilon$ in space and time. This can be seen as a model for the spatial dynamics of a…
We consider an infinite system of particles in one dimension, each particle performs independant Sinai's random walk in random environment. Considering an instant $t$, large enough, we prove a result in probability showing that the…