Equivalence between the Abelian sandpile model and the q→0 limit of the Potts model

Abstract

We establish an equivalence between the undirected Abelian sandpile model and the q→0 limit of the q-state Potts model. The equivalence is valid for arbitrary finite graphs. Two-dimensional Abelian sandpile models, thus, correspond to a conformal field theory with central charge c = -2. The equivalence also gives a Monte Carlo algorithm to generate random spanning trees. We study the growth process of the spread of fire under the burning algorithm in the background of a random recurrent configuration of the Abelian sandpile model. The average number of sites burnt upto time t varies at t^a. In two dimensions our numerically determined value of a agrees with the theoretical prediction a=8/5 . We relate this exponent to the conventional exponents characterizing the distributions of avalanche sizes.