Boundary effects in a two-dimensional Abelian sandpile

(Received 22 March 1993, accepted 25 April 1993)

Abstract
We study boundary and finite-size effects in the Abelian sandpile model due to Bak, Tang and Wiesenfeld. In the case of half-plane geometry, the probability of a unit height at the boundary, and at a distance r inside the sample is found for open and closed boundary conditions. The leading asymptotic form of the correlation functions for the unit heights, , in the strip and half-plane geometries is obtained for different boundary conditions too. Our results confirm the hypothesis that the unit height behaves like the local energy operator in the zero-component limit of the Potts model.