Related papers: On the range of the transient frog model on Z
The diffusion of a particle in a crowded environment typically proceeds through three regimes: for very short times the particle diffuses freely until it collides with an obstacle for the first time, while for very long times diffusion the…
The diffusion equation and its time-fractional counterpart can be obtained via the diffusion limit of continuous-time random walks with exponential and heavy-tailed waiting time distributions. The space dependent variable-order…
We study the asymptotic behavior of the simple random walk on oriented versions of $\mathbb{Z}^2$. The considered lattices are not directed on the vertical axis but unidirectional on the horizontal one, with random orientations whose…
We study a class of discrete-time random walks in $\mathbb{R}^d$ whose conditional drift decays polynomially in time and grows polynomially with the distance from the origin to the current position. This class is related to several models…
We study models of continuous time, symmetric, $\Z^d$-valued random walks in random environments. One of our aims is to derive estimates on the decay of transition probabilities in a case where a uniform ellipticity assumption is absent. We…
We consider random interlacements on Z^d, with d bigger or equal to 3, when their vacant set is in a strongly percolative regime. We derive an asymptotic upper bound on the probability that the random interlacements disconnect a box of…
Let $S_n$ be the simple random walk on the integer lattice $\mathbb{Z}^d$. For a Bernstein function $\phi$ we consider a random walk $S^\phi_n$ which is subordinated to $S_n$. Under a certain assumption on the behaviour of $\phi$ at zero we…
We analyse how simple local constraints in two dimensions lead a defect to exhibit robust, non-transient, and tunable, subdiffusion. We uncover a rich dynamical phenomenology realised in ice- and dimer-type models. On the microscopic scale…
A random walk on a countable group $G$ acting on a metric space $X$ gives a characteristic called the drift which depends only on the transition probability measure $\mu$ of the random walk. The drift is the `translation distance' of the…
We introduce the concept of a deterministic walk in a deterministic environment on a countable state space (DWDE). For the deterministic walk in a fixed environment we establish properties analogous to those found in Markov chain theory,…
We consider a one dimensional random walk in random environment that is uniformly biased to one direction. In addition to the transition probability, the jump rate of the random walk is assumed to be spatially inhomogeneous and random. We…
Consider an arbitrary transient random walk on $\Z^d$ with $d\in\N$. Pick $\alpha\in[0,\infty)$ and let $L_n(\alpha)$ be the spatial sum of the $\alpha$-th power of the $n$-step local times of the walk. Hence, $L_n(0)$ is the range,…
We prove that the probability the frog model with death and drift on the $d$-ary tree is recurrent can be made positive and thus is not monotone in the drift parameter.
A zero-one law of Engelbert--Schmidt type is proven for the norm process of a transient random walk. An invariance principle for random walk local times and a limit version of Jeulin's lemma play key roles.
Mott variable range hopping is a fundamental mechanism for low-temperature electron conduction in disordered solids in the regime of Anderson localization. In a mean field approximation, it reduces to a random walk (shortly, Mott random…
We introduce a discrete-time random walk model on a one-dimensional lattice with a nonconstant sojourn time and prove that the discrete density converges to a solution of a continuum diffusion equation. Our random walk model is not…
We study random walks on the integers driven by a sample of time-dependent nearest-neighbor conductances that are bounded but are permitted to vanish over time intervals of positive Lebesgue-length. Assuming only ergodicity of the…
The integer points (sites) of the real line are marked by the positions of a standard random walk. We say that the set of marked sites is weakly, moderately or strongly sparse depending on whether the jumps of the standard random walk are…
This paper investigates L\'evy walks with random velocities, extending classical models beyond constant speed assumptions. We derive scaling limits, demonstrating that diffusion depends on interplay between heavy-tailed duration and…
We consider a walker that at each step keeps the same direction with a probabilitythat depends on the time already spent in the direction the walker is currently moving. In this paper, we study some asymptotic properties of this persistent…