Related papers: Variable range random walk
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 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…
We consider the distribution of the duration time, the time elapsed since it began, of a diffusion process given its present position, under the assumption that the process began at the origin. For unbiased diffusion, the distribution does…
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…
We obtain sharp upper and lower bounds for the moderate deviations of the volume of the range of a random walk in dimension five and larger. Our results encompass two regimes: a Gaussian regime for small deviations, and a stretched…
Random walk has wide applications in many fields, such as machine learning, biology, physics, and chemistry. Random walk can be discrete or continuous in time and space. Asymmetric random walk could be described by drift-diffusion equation.…
We introduce the pushy random walk, where a walker can push multiple obstacles, thereby penetrating large distances in environments with finite obstacle density. This process provides a minimal model for experimentally observed interactions…
We investigate fluctuation phenomena for the graph distance and the number of cut points associated with random media arising from the range of a random walk. Our results demonstrate a sequence of dimension-dependent phase transitions in…
Random walks as well as diffusions in random media are considered. Methods are developed that allow one to establish large deviation results for both the `quenched' and the `averaged' case.
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 analyze random walk through fractal environments, embedded in 3-dimensional, permeable space. Particles travel freely and are scattered off into random directions when they hit the fractal. The statistical distribution of the flight…
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…
We survey recent results of normal and anomalous diffusion of two types of random motions with long memory in ${\Bbb R}^d$ or ${\Bbb Z}^d$. The first class consists of random walks on ${\Bbb Z}^d$ in divergence-free random drift field,…
We introduce a multidimensional walk with memory and random tendency. The asymptotic behaviour is characterized, proving a law of large numbers and showing a phase transition from diffusive to superdiffusive regimes. In first case, we…
In this work we establish a link between two different phenomena that were studied in a large and growing number of biological, composite and soft media: the diffusion in compartmentalized environment and the Brownian yet non-Gaussian…
We consider a random walk model in a one-dimensional environment, formed by several zones of finite width with the fixed transition probabilities. It is also assumed that the transitions to the left and right neighboring points have unequal…
We consider conservative cross-diffusion systems for two species where individual motion rates depend linearly on the local density of the other species. We develop duality estimates and obtain stability and approximation results. We first…
We prove that random walks in random environments, that are exponentially mixing in space and time, are almost surely diffusive, in the sense that their scaling limit is given by the Wiener measure.
In this paper we study a random walk in a one-dimensional dynamic random environment consisting of a collection of independent particles performing simple symmetric random walks in a Poisson equilibrium with density $\rho \in (0,\infty)$.…
We analyse the mixing profile of a random walk on a dynamic random permutation, focusing on the regime where the walk evolves much faster than the permutation. Two types of dynamics generated by random transpositions are considered: one…