Related papers: Critical branching random walk in an IID environme…

We propose a model of a one-dimensional random walk in dynamic random environment that interpolates between two classical settings: (I) the random environment is sampled at time zero only; (II) the random environment is resampled at every…

We consider a random walk on $\Z$ that branches at the origin only. In the supercritical regime we establish a law of large number for the maximal position $M_n$. Then we determine all possible limiting law for the sequence $M_n -\alpha n$…

We introduce the notion of \emph{localization at the boundary} for conditioned random walks in i.i.d. and uniformly elliptic random environment on $\mathbb{Z}^d$, in dimensions two and higher. Informally, this means that the walk spends a…

We study the simple random walk dynamics on an annealed version of a Small-World Network (SWN) consisting of $N$ nodes. This is done by calculating the mean number of distinct sites visited S(n) and the return probability $P_{00}(t)$ as a…

We investigate a branching random walk where the displacements are independent from the branching mechanism and have a stretched exponential distribution. We describe the positions of the particles in the vicinity of the rightmost particle…

We study the total mass of the solution to the parabolic Anderson model on a regular tree with an i.i.d. random potential whose marginal distribution is double-exponential. In earlier work we identified two terms in the asymptotic expansion…

Place an obstacle with probability $1-p$ independently at each vertex of $\mathbb Z^d$, and run a simple random walk until hitting one of the obstacles. For $d\geq 2$ and $p$ strictly above the critical threshold for site percolation, we…

We consider a system of $N$ particles on the real line that evolves through iteration of the following steps: 1) every particle splits into two, 2) each particle jumps according to a prescribed displacement distribution supported on the…

We consider branching random walks in $d$-dimensional integer lattice with time-space i.i.d. offspring distributions. This model is known to exhibit a phase transition: If $d \ge 3$ and the environment is "not too random", then, the total…

We consider branching random walks in $d$-dimensional integer lattice with time-space i.i.d. offspring distributions. When $d \ge 3$ and the fluctuation of the environment is well moderated by the random walk, we prove a central limit…

We consider Activated Random Walks on $\Z$ with totally asymmetric jumps and critical particle density, with different time scales for the progressive release of particles and the dissipation dynamics. We show that the cumulative flow of…

We study a $d$-dimensional branching random walk (BRW) in an i.i.d. random environment on $\mathbb{Z}^d$ in discrete time. A Bernoulli trap field is attached to $\mathbb{Z}^d$, where each site, independently of the others, is a trap with a…

We study a branching random walk on $\r$ with an absorbing barrier. The position of the barrier depends on the generation. In each generation, only the individuals born below the barrier survive and reproduce. Given a reproduction law,…

A critical branching process $\left\{ Z_{k},k=0,1,2,...\right\} $ in a random environment is considered. A conditional functional limit theorem for the properly scaled process $\left\{ \log Z_{pu},0\leq u<\infty \right\} $ is established…

We consider a modification of classical branching random walk, where we add i.i.d. perturbations to the positions of the particles in each generation. In this model, which was introduced and studied by Bandyopadhyay and Ghosh (2023),…

We develop a systematic analytic approach to the problem of branching and annihilating random walks, equivalent to the diffusion-limited reaction processes 2A->0 and A->(m+1)A, where m>=1. Starting from the master equation, a…

We study the quenched behaviour of a perturbed version of the simple symmetric random walk on the set of integers. The random walker moves symmetrically with an exception of some randomly chosen sites where we impose a random drift. We show…

For a branching process in random environment it is assumed that the offspring distribution of the individuals varies in a random fashion, independently from one generation to the other. Interestingly there is the possibility that the…

Let $\mathcal{T}$ be a supercritical Galton-Watson tree with a bounded offspring distribution that has mean $\mu >1$, conditioned to survive. Let $\varphi_{\mathcal{T}}$ be a random embedding of $\mathcal{T}$ into $\mathbb{Z}^d$ according…

Via a Dirichlet form extension theorem and making full use of two-sided heat kernel estimates, we establish quenched invariance principles for random walks in random environments with a boundary. In particular, we prove that the random walk…