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 ρ ∈ (0, ∞). At each step the random walk performs a nearest-neighbour jump, moving to the right with probability p • when it is on a vacant site and probability p • when it is on an occupied site. Assuming that p • ∈ (0, 1) and p • = 1 2 , we show that the position of the random walk satisfies a strong law of large numbers, a functional central limit theorem and a large deviation bound, provided ρ is large enough. The proof is based on the construction of a renewal structure together with a multiscale renormalisation argument.MSC 2010. Primary 60F15, 60K35, 60K37; Secondary 82B41, 82C22, 82C44.
A solid wooden cube fragments into pieces as we sequentially drill holes through it randomly. This seemingly straightforward observation encompasses deep and nontrivial geometrical and probabilistic behavior that is discussed here. Combining numerical simulations and rigorous results, we find off-critical scale-free behavior and a continuous transition at a critical density of holes that significantly differs from classical percolation.
We introduce a percolation model on Z d , d ≥ 3, in which the discrete lines of vertices that are parallel to the coordinate axis are entirely removed at random and independently of each other. In this way a vertex belongs to the vacant set V if and only if none of the d lines to which it belongs, is removed. We show the existence of a phase transition for V as the probability of removing the lines is varied. We also establish that, in the certain region of parameters space where V contains an infinite component, the truncated connectivity function has power-law decay, while inside the region where V has no infinite component, there is a transition from exponential to power-law decay. In the particular case d = 3 the power-law decay extends through all the region where V has an infinite connected component. We also show that the number of infinite connected components of V is either 0, 1 or ∞. April 2, 2018. arXiv:1509.06204v1 [math.PR] 21 Sep 2015
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