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

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.