Abstract. We study the hydrodynamic and the hydrostatic behavior of the Simple Symmetric Exclusion Process with slow boundary. The term slow boundary means that particles can be born or die at the boundary sites, at a rate proportional to N −θ , where θ > 0 and N is the scaling parameter. In the bulk, the particles exchange rate is equal to 1. In the hydrostatic scenario, we obtain three different linear profiles, depending on the value of the parameter θ; in the hydrodynamic scenario, we obtain that the time evolution of the spatial density of particles, in the diffusive scaling, is given by the weak solution of the heat equation, with boundary conditions that depend on θ. If θ ∈ (0, 1), we get Dirichlet boundary conditions, (which is the same behavior if θ = 0, see [7]); if θ = 1, we get Robin boundary conditions; and, if θ ∈ (1, ∞), we get Neumann boundary conditions.

In this paper we study noise sensitivity and threshold phenomena for Poisson Voronoi percolation on R 2 . In the setting of Boolean functions, both threshold phenomena and noise sensitivity can be understood via the study of randomized algorithms. Together with a simple discretization argument, such techniques apply also to the continuum setting. Via the study of a suitable algorithm we show that boxcrossing events in Voronoi percolation are noise sensitive and present a threshold phenomenon with polynomial window. We also study the effect of other kinds of perturbations, and emphasize the fact that the techniques we use apply for a broad range of models.

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We study a detection problem in the following setting: On the one-dimensional integer lattice, at time zero, place nodes on each site independently with probability ρ ∈ [0, 1) and let them evolve as a simple symmetric exclusion process. At time zero, place a target at the origin. The target moves only at integer times, and can move to any site that is within distance R from its current position. Assume also that the target can predict the future movement of all nodes. We prove that, for R large enough (depending on the value of ρ) it is possible for the target to avoid detection forever with positive probability. The proof of this result uses two ingredients of independent interest. First we establish a renormalisation scheme that can be used to prove percolation for dependent oriented models under a certain decoupling condition. This result is general and does not rely on the specifities of the model. As an application, we prove our main theorem for different dynamics, such as independent random walks and independent renewal chains. We also proof existence of oriented percolation for random interlacements and for its vacant set for large dimensions . The second step of the proof is a space-time decoupling for the exclusion process. *

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