Thursday, September 24, 2015 - 15:05
1 hour (actually 50 minutes)
The Abelian sandpile was invented as a "self-organized critical" model whose stationary behavior is similar to that of a classical statistical mechanical system at a critical point. On the d-dimensional lattice, many variables measuring correlations in the sandpile are expected to exhibit power-law decay. Among these are various measures of the size of an avalanche when a grain is added at stationarity: the probability that a particular site topples in an avalanche, the diameter of an avalanche, and the number of sites toppled in an avalanche. Various predictions about these exist in the physics literature, but relatively little is known rigorously. We provide some power-law upper and lower bounds for these avalanche size variables and a new approach to the question of stabilizability in two dimensions.