Hayk Aleksanyan: Discrete Balayage and Boundary Sandpile

Motivated by a connection between Laplacian growth models and quadrature domains, we introduce a new growth model on the integer lattice which we named boundary sandpile, in our search to model a quadrature surface by particle (sandpile) dynamics. Our model amounts to potential theoretic redistribution of the given mass distribution on the lattice onto the discrete boundary of an apriori unknown domain. In this talk we will discuss the background and motivation behind the model, and will discuss some basic properties of it. 

As an application of our methods developed for the boundary sandpile, combined with the least action principle of Fey, Levine, and Peres, we will prove that the free boundary of the scaling limit of the celebrated Abelian Sandpile model (ASM) is locally a Lipschitz graph; this we believe is the first instance addressing the qualitative properties of the boundary of ASM.

Our analysis is largely motivated by potential theory and free boundary problems, and builds on top of these their combinatorial and probabilistic (random walk) counterparts.

The talk is based on a joint work with Henrik Shahgholian.