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Phase Transitions and Near-Critical Phenomena in the Abelian Sandpile Model

Author
McDermott, Ryan Keith
Abstract
In chapter 2 we investigate the behavior around the fixed-energy sandpile's phase transition as conjectured in [9]. In the course of our investigations we define a supercritical threshold density and a subcritical threshold density based upon conditions of the sandpile odometers, and demonstrate the retention and loss of sand beyond each of these threshold densities, respectively. We conjecture the equivalence of these densities, and give a partial proof in d=1. In chapter 3 we obtain the scaling limit of the odometer functions of the sandpile on the integers having 2 chips with probability p, and 0 chips with probability 1-p, as we send p to 1/2, the critical value, from below. In chapter 4 we extend Lionel Levine's findings in [12] to the case of supercritical sandpiles with initial height tending to infinity.
Description
78 pages
Date Issued
2021-08Committee Chair
Levine, Lionel
Committee Member
Sosoe, Philippe; Saloff-Coste, Laurent Pascal
Degree Discipline
Mathematics
Degree Name
Ph. D., Mathematics
Degree Level
Doctor of Philosophy
Type
dissertation or thesis