It is well-known from work by Kesten in the mid-eighties and work by Borgs, Chayes, Kesten and Spencer around 2000 that, informally speaking, the largest, the second largest, the third largest etcetera cluster in an $n$ times $n$ box in 2D critical percolation typically have size of order $n^2 \pi(n)$. Here $\pi(n)$ is the probability that a given vertex v has an open path to vertices at distance at least $n$ from v. Motivated by so-called frozen percolation problems (which I will briefly explain), we obtained some new modifications/refinements of these results. I will also address our work on an open problem in this area posed in a paper by Jarai (2003). Part of this talk is based on joint work with R. Conijn and D. Kiss.

The BK inequality says, roughly speaking, that the probability that two events `occur disjointly' is smaller than or equal to the product of the two individual probabilities. It has been used fruitfully in percolation and related topics. Until recently, no inequalities of this type were known for situations where the underlying random variables are dependent. However, in the last two years considerable progress has been made in this direction. This talk will start with an introduction and then highlight these new developments. Partly based on joint work with Johan Jonasson and joint work with Alberto Gandolfi.