Related papers: Maximal displacement in a branching random walk th…
We are interested in the biased random walk on a supercritical Galton--Watson tree in the sense of Lyons, Pemantle and Peres, and study a phenomenon of slow movement. In order to observe such a slow movement, the bias needs to be random;…
We study a symmetric random walk (RW) in one spatial dimension in environment, formed by several zones of finite width, where the probability of transition between two neighboring points and corresponding diffusion coefficient are…
A deterministic walk in a random environment can be understood as a general random process with finite-range dependence that starts repeating a loop once it reaches a site it has visited before. Such process lacks the Markov property. We…
We study branching random walks in random environment on the $d$-dimensional square lattice, $d \geq 1$. In this model, the environment has finite range dependence, and the population size cannot decrease. We prove limit theorems (laws of…
We consider one-dimensional branching Brownian motion in spatially random branching environment (BBMRE) and show that for almost every realisation of the environment, the distributions of the maximal particle of the BBMRE re-centred around…
We consider a simple model for active random walk with general temporal correlations, and investigate the shape of the probability distribution function of the displacement during a short time interval. We find that under certain conditions…
The kinetics of a variety of transport-controlled processes can be reduced to the problem of determining the mean time needed to arrive at a given location for the first time, the so called mean first passage time (MFPT) problem. The…
We consider the long-time behaviour of a branching random walk in random environment on the lattice $\Z^d$. The migration of particles proceeds according to simple random walk in continuous time, while the medium is given as a random…
We study persistent random walk with time dependent velocity reversal probabilities and identify a criterion for a non-equilibrium dynamical transition. As a representative example, we consider a power law reversal probability $p(t)\sim…
When navigating complex environments, animals often combine multiple strategies to mitigate the effects of external disturbances. These modalities often correspond to different sources of information, leading to speed-accuracy trade-offs.…
We consider a system of independent one-dimensional random walks in a common random environment under the condition that the random walks are transient with positive speed $v_P$. We give upper bounds on the quenched probability that at…
In this paper, we investigate random walks in a family of small-world trees having an exponential degree distribution. First, we address a trapping problem, that is, a particular case of random walks with an immobile trap located at the…
We study the random walk of a particle in a compartmentalized environment, as realized in biological samples or solid state compounds. Each compartment is characterized by its length $L$ and the boundaries transmittance $T$. We identify two…
The random walk process in a nonhomogeneous medium, characterised by a L\'evy stable distribution of jump length, is discussed. The width depends on a position: either before the jump or after that. In the latter case, the density slope is…
We consider a critical branching process in an i.i.d. random environment, in which one immigrant arrives at each generation. We are interested in the event $\mathcal{A}_i(n)$ that all individuals alive at time $n$ are offspring of the…
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…
Many systems across the sciences evolve through a combination of multiplicative growth and diffusive transport. In the presence of disorder, these systems tend to form localized structures which alternate between long periods of relative…
We consider a branching random walk initiated by a single particle at location 0 in which particles alternately reproduce according to the law of a Galton-Watson process and disperse according to the law of a driftless random walk on the…
Random walks find applications in many areas of science and are the heart of essential network analytic tools. When defined on temporal networks, even basic random walk models may exhibit a rich spectrum of behaviours, due to the…
In this paper, we study a spatial model for dormancy in random environment via a two-type branching random walk in continuous-time, where individuals can switch between dormant and active states through spontaneous switching independent of…