Limit shapes in the Abelian Sandpile Model

 In a series of 2-3 talks we will discuss the abelian sandpile
model (ASM for short). The ASM is a growth model in which grains of sand
are placed at the vertices of a graph, and they spread according to a
toppling rule. A site topples if its amount of sand is at least as large
as the degree of the underlying vertex, in which case it sends one grain
of sand to each neighbor.

The first talks will consist of an informal introduction to the model,
with a discussion of a few interesting open questions related to the model.

In the last talk we will discuss recent results about a dissipative
version of the model, called the Leaky-ASM, in which in each topple a
site sends some sand to each neighbor and leaks a portion 1-1/d of its
sand. This is a dissipative generalization of the Abelian Sandpile
Model, which corresponds to the case d=1.

We will discuss how, by connecting the model to a certain killed random
walk on the square lattice, for any fixed d>1, an explicit limit shape
can be computed for the region visited by the sandpile when it stabilizes.

We will also discuss the limit shape in the regime when the dissipation
parameter d converges to 1 as n grows, as this is related to the
ordinary ASM with a modified initial configuration.

Monday, October 17, 2022 - 09:30

via Microsoft Teams. If interested in attending, please email:

Speaker Information
Sevak Mkrtchyan
University of Rochester