# Boundary conditions in Abelian sandpiles

• Samuel Gamlin

Student thesis: Doctoral ThesisPhD

### Abstract

The focus of this thesis is to investigate the impact of the boundary conditions onconfigurations in the Abelian sandpile model. We have two main results to present inthis thesis.Firstly we give a family of continuous, measure preserving, almost one-to-one mappingsfrom the wired spanning forest to recurrent sandpiles. In the special case of $Z^d$,$d \geq 2$, we show how these bijections yield a power law upper bound on the rate ofconvergence to the sandpile measure along any exhaustion of $Z^d$.Secondly we consider the Abelian sandpile on ladder graphs. For the ladder sandpilemeasure, $\nu$, a recurrent configuration on the boundary, I, and a cylinder event, E, weprovide an upper bound for$\nu(E|I) − \nu(E)$.
Date of Award 1 Apr 2016 English University of Bath Antal Jarai (Supervisor)

### Keywords

• Abelian sandpile
• spanning forest
• wilson's algorithm
• Random walk

### Cite this

Boundary conditions in Abelian sandpiles
Gamlin, S. (Author). 1 Apr 2016

Student thesis: Doctoral ThesisPhD