We compare several growth models on the two dimensional lattice. In some models, like internal DLA and rotor-router aggregation, the scaling limits are universal; in particular, starting from a point source yields a disk. In the abelian sandpile, particles are added at the origin and whenever a site has four particles or more, the top four particles topple, with one going to each neighbor. Despite similarities to other models, for the sandpile, the intriguing pattern that arises is not circular and depends on the particular lattice. A scaling limit exists for the sandpile, as was recently shown by Pegden and Smart, but it is not universal and is still mysterious. This research has been greatly influenced by pictures of the relevant sets, which I will show in the talk. They suggest a connection to conformal mapping which has not been established yet. (Talk based on joint works with Lionel Levine.)
Following Monday's lecture, there will be a reception in honor of Professor Peres at the Koenig Alumni Center, 1202 University Avenue (the SE corner of Broadway and University).
Laplacian Growth and the Mystery of the Abelian Sandpile: a Visual Tour
Laplacian Growth and the Mystery of the Abelian Sandpile: a Visual Tour