We review the status of the two-dimensional Abelian sandpile model as a strong candidate to provide a lattice realization of logarithmic conformal invariance with central charge c = −2. Evidence supporting this view is collected from various aspects of the model. These include the study of some conformally invariant boundary conditions, and the corresponding boundary condition changing fields, the calculation of correlations of certain bulk and boundary observables (the height variables) as well as a proper account of the necessary dissipation, which allows for a physical understanding of some of the strange but generic features of logarithmic theories.a finite system, this measure is simple but non-local with respect to the natural degrees of freedom, namely the heights of the pile at each site, which are strongly correlated throughout the lattice. Such intrinsically non-local features are a landmark of LCFTs, and qualifies the ASM as a good candidate for an LCFT description. Indeed most explicit computations to date aim at providing evidence that the scaling limit of the measure P is the field-theoretical measure of an LCFT, with central charge c = −2.The rest of the article will be devoted to review the exact results which are the most convincing to support the previous assertion. Such calculations include the calculation of certain bulk correlators, the discussion of a few boundary conditions and the corresponding boundary condition changing fields, and some boundary and bulk correlators, some of them forming Jordan cells. In the ASM, an absolutely crucial rôle is played by dissipation, which is essential to make the dynamics well-defined. In the conformal picture, the insertion of dissipation is represented by a dimension 0 field, logarithmic partner of the identity, which allows for a transparent understanding of some of the strangest features of LCFT.We should mention that the ASM is not the only lattice model believed to be described by a logarithmic conformal theory with c = −2. At least two other models are known, namely the dense polymer model which, in its loop formulation, is the first of the infinite series of so-called logarithmic minimal models [2], and the dimer model [3]. In their scaling limit, the three models are believed to be different. Although very close and even equivalent in certain instances, the ASM and the dimer model are not when the lattice has one or more periodic direction (see below in Section 2.4). Some differences between the dense polymer model and the dimer model have been recently emphasized in [4]. Finally, while the dense polymer is likely to be described by the symplectic free fermion theory, it is not the case of the ASM, as an argument recalled in Section 5.1 shows. Exactly which conformal theories describe the dimer model and the ASM is a widely open question.
The Abelian sandpile modelWe review in this Section the definition of the model and its basic features, omitting most of the time the detailed proofs. These and more details about the model can be found f.i. in ...
