Mathematics Colloquium

Percolation is a spatial stochastic process, introduced long ago by Hammersley and Welsh to model the flow of fluid through a porous medium. The object of study is a random subgraph of the lattice Z^d with fixed density. In spite of this simple definition,the model has an intricate mathematical structure, and its study has led to many deep discoveries of importance in other fields of mathematics, notably Schramm's Stochastic Loewner Evolution. In this talk, I will concentrate on one aspect of percolation, the "chemical distance", that is the graph distance inside components of the percolation subgraph. While in some cases (the "supercritical phase"), this random quantity has a satisfactory description, in the most interesting ("critical") case, it remains a mystery, and SLE seems to be of no help. I will talk about the few results that are known, including some recent ones with Jack Hanson and Michael Damron.