Honors Oral Exam

The Limit Shape of the Leaky Abelian Sandpile Model with Multiple Starting Points.

Lloyd Page (University of Rochester)

Tuesday, April 26th, 2022
9:30 AM - 10:20 AM
Hutchison 473 (with possible livestream on zoom 986 2892 7517)

The leaky abelian sandpile model with multiple starting points is a growth model in which n grains of sand are started at either a finite number of starting points or a configuration of starting points dependent on n in \(\mathbb{Z}^2\) . These grains of sand spread out along the vertices of \(\mathbb{Z}^2\) according to a toppling rule. A site at a vertex topples if the number of grains of sand at said vertex are above a specified threshold. In such a toppling, the site sends some sand to each of its neighbors and leaks a portion, \(1-\frac{1}{d}\) , of the toppled sand. A site may topple multiple times before it falls below the threshold and stops toppling. I explored the limit shape in the symmetric case with more than 1 source point. In this case, each topple sends an equal amount of sand to each neighbor. Supposing we have k source points where k is finite, I show that the limit shape is the union of k limit shapes, each originating at one of those source points. This means that as \(d \to 1\), we have a union of circles, and as \(d \to \infty\), we have a union of diamonds. I also explored a starting arrangement of a square centered at the origin with log n length sides. However, I was unable to prove any significant conclusions about the limit shape in this case.

Event contact: jonathan dot pakianathan at rochester dot edu