Related papers: Using Random Walks to Establish Wavelike Behavior …

The random walk with hyperbolic probabilities that we are introducing is an example of stochastic diffusion in a one-dimensional heterogeneous media. Although driven by site-dependent one-step transition probabilities, the process retains…

Continuous time random Walk model has been versatile analytical formalism for studying and modeling diffusion processes in heterogeneous structures, such as disordered or porous media. We are studying the continuous limits of Heterogeneous…

We consider random walks on marked simple point processes with symmetric jump rates and unbounded jump range. We prove homogenization properties of the associated Markov generators. As an application, we derive the hydrodynamic limit of the…

We study the problem of homogenization for inertial particles moving in a time dependent random velocity field and subject to molecular diffusion. We show that, under appropriate assumptions on the velocity field, the large--scale,…

We study a random walk in random environment on the non-negative integers. The random environment is not homogeneous in law, but is a mixture of two kinds of site, one in asymptotically vanishing proportion. The two kinds of site are (i)…

In this paper we study extreme events for random walks on homogeneous spaces. We consider the following three cases. On the torus we study closest returns of a random walk to a fixed point in the space. For a random walk on the space of…

We prove a law of large numbers for random walks in certain kinds of i.i.d. random environments in Z^d that is an extension of a result of Bolthausen, Sznitman and Zeitouni (2003). We use this result, along with the lace expansion for…

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 a population of $N$ labeled random walkers moving on a substrate, and an excitation jumping among the walkers upon contact. The label $\mathcal{X}(t)$ of the walker carrying the excitation at time $t$ can be viewed as a…

We investigate the dynamics of the Fermi--Pasta--Ulam--Tsingou chain with long-wavelength random initial data. When the energy per particle is small, thermal equilibrium is not reached on a fast timescale and the system enters…

Random forests are a statistical learning method widely used in many areas of scientific research because of its ability to learn complex relationships between input and output variables and also its capacity to handle high-dimensional…

Let $G$ be a finitely generated group of polynomial volume growth equipped with a word-length $|\cdot|$. The goal of this paper is to develop techniques to study the behavior of random walks driven by symmetric measures $\mu$ such that, for…

We investigate the time-evolution of elastoplastic materials reinforced by randomly distributed long-range interactions. Starting from a rate-independent system on a discrete spring lattice that combines local linearized elasticity,…

A comparison technique for finite random walks on finite graphs is introduced, using the well-known interlacing method. It yields improved return probability bounds. A key feature is the incorporation of parts of the spectrum of the…

Let $\tau = (\tau_i : i \in {\Bbb Z})$ denote i.i.d.~positive random variables with common distribution $F$ and (conditional on $\tau$) let $X = (X_t : t\geq0, X_0=0)$, be a continuous-time simple symmetric random walk on ${\Bbb Z}$ with…

Quantum and random walks have been shown to be equivalent in the following sense: a time-dependent random walk can be constructed such that its vertex distribution at all time instants is identical to the vertex distribution of any…

We consider random walks in a uniformly elliptic, balanced, i.i.d. random environment in the integer lattice $Z^d$ for $d\geq 2$ and the corresponding problem of stochastic homogenization of non-divergence form difference operators. We…

We address the homogenization of a semilinear hyperbolic stochastic partial differential equation with highly oscillating coefficients, in the context of ergodic algebras with mean value. To achieve our goal, we use a suitable variant of…

On infinite homogeneous structures, two random walkers meet with certainty if and only if the structure is recurrent, i.e., a single random walker returns to its starting point with probability 1. However, on general inhomogeneous…

As an example for complex systems with extreme events we investigate ocean wave states exhibiting rogue waves. We present a statistical method of data analysis based on multi-point statistics which for the first time allows grasping extreme…