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We consider the unoriented two-dimensional Abelian sandpile model from a perspective based on two-dimensional (conformal) field theory. We compute lattice correlation functions for various cluster variables (at and off criticality), from which we infer the field-theoretic description in the scaling limit. We find perfect agreement with the predictions of a c=-2 conformal field theory and its massive perturbation, thereby providing direct evidence for conformal invariance and more generally for a description in terms of a local field theory. The question of the height 2 variable is also addressed, with, however, no definite conclusion yet.

We compute the lattice 1-site probabilities, on the upper half-plane, of the four height variables in the two-dimensional Abelian sandpile model. We find their exact scaling form when the insertion point is far from the boundary, and when the boundary is either open or closed. Comparing with the predictions of a logarithmic conformal theory with central charge c = −2, we find a full compatibility with the following field assignments: the heights 2, 3 and 4 behave like (an unusual realization of) the logarithmic partner of a primary field with scaling dimension 2, the primary field itself being associated with the height 1 variable. Finite size corrections are also computed and successfully compared with numerical simulations. Relying on these field assignments, we formulate a conjecture for the scaling form of the lattice 2-point correlations of the height variables on the plane, which remain as yet unknown. The way conformal invariance is realized in this system points to a local field theory with c = −2 which is different from the triplet theory. PACS numbers: 05.65.+b,45.70.-n I. INTRODUCTION Sandpile models are open dynamical systems which generically show a very rich spectrum of critical behaviours, both spatially and temporally. Along with many other models, they are complex critical models. Because of their non-linear dynamics and their non-local features, a fixed external driving can trigger events whose scales follow critical distributions. The references [1-3] review the ideas and models involved.One of the simplest and yet most challenging models is the two-dimensional model originally proposed by Bak, Tang and Wiesenfeld [4], now referred to as the BTW model, or the unoriented Abelian sandpile model (ASM) after it was shown by Dhar to possess an Abelian group [5]. It is this model we consider in this article.The most natural formulation of the ASM is in terms of discrete height variables, taking the four integral values 1 to 4 and located at the sites of a grid. The configuration of the sandpile,

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We consider the unoriented two-dimensional Abelian sandpile model on the half-plane with open and closed boundary conditions. We show that the operator effecting the change from closed to open, or from open to closed, is a boundary primary field of weight −1/8, belonging to a c = −2 logarithmic conformal field theory.