Abstract

First-passage percolation (FPP), introduced by Hammersley and Welsh in the 60s, is a model that describes fluid flow in a porous medium. Mathematically, one considers the integer lattice with nearest neighbor edges, puts i.i.d. nonnegative weights on the edges, and studies the induced (pseudo)metric. Our main goal is to understand the large scale geometry of this random metric space. In the first talk, we will study the asymptotic behavior of a metric ball as the radius tends to infinity. We prove the so-called “shape theorem”, which tells us that a large ball, after rescaled properly, will tend to a deterministic limit shape. If time allows, we will also discuss some known results about the limit shape, and talk about some related open problems.