The focus of this thesis is to investigate the impact of the boundary conditions on configurations in the Abelian sandpile model. We have two main results to present in this thesis. Firstly we give a family of continuous, measure preserving, almost one-to-one mappings from 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 of convergence to the sandpile measure along any exhaustion of $Z^d$. Secondly we consider the Abelian sandpile on ladder graphs. For the ladder sandpile measure, $\nu$, a recurrent configuration on the boundary, I, and a cylinder event, E, we provide an upper bound for $\nu(E|I) − \nu(E)$.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:687371 |
| Date | January 2016 |
| Creators | Gamlin, Samuel |
| Contributors | Jarai, Antal |
| Publisher | University of Bath |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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