Marcelo Hilario (UFMG)

Consider the hyper-cubic lattice and remove the lines parallel to the coordinate axis independently at random. Does the set of remaining vertices undergo a sharp phase transition as the probability of removing the lines vary? How many connected components are there? In this talk we discuss these question for this model and for a continuous analogous model in which we remove cylinders from the Euclidian space in a isometry invariant way. We also discuss for Bernoulli bond percolation processes in the square lattice, how enhancing the parameter in a set of vertical lines chosen uniformly at random changes the critical point.