### Random walk attracted by percolation clusters.

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We consider the continuous time, one-dimensional random walk in random environment in Sinai’s regime. We show that the probability for the particle to be, at time $t$ and in a typical environment, at a distance larger than ${t}^{a}$ ($0\<a\<1$) from its initial position, is $exp\{-\mathrm{Const}\xb7{t}^{a}/[(1-a)lnt](1+o\left(1\right))\}$.

We consider a random walk in a stationary ergodic environment in $\mathbb{Z}$, with unbounded jumps. In addition to uniform ellipticity and a bound on the tails of the possible jumps, we assume a condition of strong transience to the right which implies that there are no “traps.” We prove the law of large numbers with positive speed, as well as the ergodicity of the environment seen from the particle. Then, we consider Knudsen stochastic billiard with a drift in a random tube in ${\mathbb{R}}^{d}$, $d\ge 3$, which serves as environment....

In this paper we establish a decoupling feature of the random interlacement process ${\mathcal{I}}^{u}\subset {\mathbb{Z}}^{d}$ at level $u$, $d\ge 3$. Roughly speaking, we show that observations of ${\mathcal{I}}^{u}$ restricted to two disjoint subsets ${A}_{1}$ and ${A}_{2}$ of ${\mathbb{Z}}^{d}$ are approximately independent, once we add a sprinkling to the process ${\mathcal{I}}^{u}$ by slightly increasing the parameter $u$. Our results differ from previous ones in that we allow the mutual distance between the sets ${A}_{1}$ and ${A}_{2}$ to be much smaller than their diameters. We then provide an important application of this...

We consider the continuous time, one-dimensional random walk in random environment in Sinai's regime. We show that the probability for the particle to be, at time and in a typical environment, at a distance larger than () from its initial position, is exp{-Const ⋅ ln(1))}.

Spider walks are systems of interacting particles. The particles move independently as long as their movements do not violate some given rules describing the relative position of the particles; moves that violate the rules are not realized. The goal of this paper is to study qualitative properties, as recurrence, transience, ergodicity, and positive rate of escape of these Markov processes.

We investigate the optimal alignment of two independent random sequences of length . We provide a polynomial lower bound for the probability of the optimal alignment to be macroscopically non-unique. We furthermore establish a connection between the transversal fluctuation and macroscopic non-uniqueness.

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