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    Plant root associated biofilms

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    The need for speed : Maximizing random walks speed on fixed environments

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    We study nearest neighbor random walks on fixed environments of Z\mathbb{Z} composed of two point types : (1/2,1/2)(1/2,1/2) and (p,1βˆ’p)(p,1-p) for p>1/2p>1/2. We show that for every environment with density of pp drifts bounded by Ξ»\lambda we have lim sup⁑nβ†’βˆžXnn≀(2pβˆ’1)Ξ»\limsup_{n\rightarrow\infty}\frac{X_n}{n}\leq (2p-1)\lambda, where XnX_n is a random walk on the environment. In addition up to some integer effect the environment which gives the best speed is given by equally spaced drifts

    Geometry of the random interlacement

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    We consider the geometry of random interlacements on the dd-dimensional lattice. We use ideas from stochastic dimension theory developed in \cite{benjamini2004geometry} to prove the following: Given that two vertices x,yx,y belong to the interlacement set, it is possible to find a path between xx and yy contained in the trace left by at most ⌈d/2βŒ‰\lceil d/2 \rceil trajectories from the underlying Poisson point process. Moreover, this result is sharp in the sense that there are pairs of points in the interlacement set which cannot be connected by a path using the traces of at most ⌈d/2βŒ‰βˆ’1\lceil d/2 \rceil-1 trajectories
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