Abstract: I will discuss recent work on the distribution of finite clusters in (slightly) supercritical percolation on nonamenable transitive graphs, and will also discuss the relevant conjectures for the high-dimensional Euclidean case. This talk will be based mostly on the papers https://arxiv.org/abs/1904.10448 and https://arxiv.org/abs/2002.02916, the first of which is joint with Jonathan Hermon.
