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.

via Microsoft Teams. If interested in attending, please email: gregconstantine314@gmail.com