Abstract: In this talk we will discuss various results related to cluster size distribution for i.i.d. bond percolation model. Our goal is to see if there is any "qualitative" difference in the tail of the distribution depending on qualitative properties of the underlying graph. We will show that for any Cayley graph, the probability (at any p) that the cluster of the origin has size n decays at a well-defined exponential rate (possibly 0). For general graphs, we relate this rate being positive in the supercritical regime with the amenability/nonamenability of the underlying graph. We will also discuss the importance of transitivity of the underlying graph. The talk will be self contained and will not assume any prior knowledge of Percolation Theory.

This is a joint work with Jeffrey Steif and Ádám Timár.