Georgia Institute of Technology College of Sciences

First of two talks.

In Bernoulli percolation, we place edges independently with probability p between vertices of a lattice, and study the resulting random geometry. In the first talk, I will give a friendly introduction to the model and discuss some of the motivating questions and the history of methods in the high dimensional setting. In the second talk, I will discuss our recently developed tool, the incipient infinite cluster, and describe how we use it to derive the exact limiting distribution of distances within clusters, as well as prove sharp asymptotics for k-point connectivity functions. These results together establish several aspects of the conjecture that the scaling limit of high dimensional critical clusters is the Integrated Super-Brownian Excursion. Based on joint work with Shirshendu Chatterjee, Jack Hanson, and Philippe Sosoe.