**Abstract:** In the usual Bernoulli site percolation on the lattice, each site is independently removed with probability .

For the Bernoulli line percolation, in contrast, entire lines of sites that are parallel to one of the coordinate axis are removed independently at random with a given probability that may depend on the direction along which the line extends. Similarly, for the Bernoulli hyperplane percolation instead of lines, sites that belong to affine hyperplanes are removed. We are interested in the features of the phase transition for the vacant set, that is, the set that has remained after the removal of the hyperplanes.

In the first half of the talk we will concentrate in the Bernoulli line percolation and show that there is a change from exponential to power-law decay for the connectivity inside some regions of the subcritical phase. We will also show power-law decay in the supercritical phase. Then we will discuss the open problem of whether the infinite cluster is unique when it exists. In particular, we will show how to prove that when an infinite cluster exists, it is either unique or there are infinitely many other clusters. In the second half of the talk we will extend the discussion for the the higher-dimensional setting showing some similar results for Bernoulli hyperplane percolation.

## Video