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Recently, Duminil-Copin, Raoufi and Tassion developed a new method to prove sharp phase transition for Voronoi percolation even in higher dimensions. The idea is based on two main steps: For $S_n(0) := \{x \in \mathbb{R}^d : ||x|| = n\}$ and $\theta_n(p) := P_p(0 \leftrightarrow S_n(0))$, they first prove a family of differential inequalities regarding $\theta_n(p)$. Here, they make use of a randomized algorithm, which determines the function $f := 1_{0 \leftrightarrow S_n(0)}$, and of the OSSS inequality, to estimate the variance of $f$. Second they employ a Lemma to $\theta_n(p)$, which verifies the sharp phase transition. In the talk we transfer this method to prove sharp phase transition for confetti percolation in $\mathbb{R}^d \times (- \infty, 0]$.
In this talk, we prove for various graphs that the random walk is recurrent or transient. While in one case the random walk almost surely visits every vertex of the graph infinitely many times, in the other case it eventually escapes any finite set of vertices and never returns. Under certain assumptions on the underlying point process, we apply results from Gurel-Gurevich, Nachmias and Rousselle to get recurrence results for graphs in the plane and transience results for higher dimensions. Apart from that we will mention some classes of point processes for which our results hold.
We consider the spread of an infection on top of a moving population. The environment evolves as a zero range process on the integer lattice starting in equilibrium. At time zero, the set of infected particles is composed by those which are on the negative axis, while particles at the right of the origin are considered healthy. A healthy particle immediately becomes infected if it shares a site with an infected particle. We prove that the front of the infection wave travels to the right with positive and finite velocity.