Physica A 185 (1992) 129-145 North-Holland Equivalence between the Abelian sandpile model and the q 0 limit of the Potts model S.N. Majumdar and Deepak Dhar Theoretical Physics Group, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India 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 ~. 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. I. Introduction In recent years the concept of self-organized criticality (SOC) has attracted much attention [1-5]. It has been found useful in the description of such diverse systems as earthquakes [6-8], forest fires [4], relaxation phenomena in magnets [9, 10] and coagulation [11]. Bak et al. in their pioneer papers [1, 2] introduced the concept through the example of sandpiles, which have been extensively studied [12-16] as paradigms of self-organized critical systems. Sandpile-like lattice models have been used to describe fracture [17], neural networks [18] and hydrogen bonding in liquid water [19]. Of special interest are the so-called Abelian sandpile models [20] (ASMs) as they provide a nontrivial, analytically tractable example of SOC. In an earlier paper [20], we have shown that if the toppling condition at a lattice site depends only on the height of sandpile at that site and not on heights at other sites, operators corresponding to sandgrain addition at different sites commute. This Abelian property was used to analytically determine the steady state of the model, and also some response functions and the spectrum of relaxation times in the critical state. While there has been much interest in the study of ASMs [21-26], so far only some of the critical exponents of interest have been determined analytical- 0378-4371/92/$05.00 © 1992-Elsevier Science Publishers B.V. All rights reserved