The four height variables, boundary correlations, and dissipative defects in the Abelian sandpile model

Abstract

We analyze the two-dimensional Abelian sandpile model, and demonstrate that the four height variables have different field identifications in the bulk, and along closed boundaries, but become identical, up to rescaling, along open boundaries. We consider two-point boundary correlations in detail, and discuss a number of complications that arise in the mapping from sandpile correlations to spanning tree correlations; the structure of our results suggests a conjecture that could greatly simplify future calculations. We find a number of three-point functions along closed boundaries, and propose closed boundary field identifications for the height variables. We analyze the effects of dissipative defect sites, at which the number of grains is not conserved, and show that dissipative defects along closed boundaries, and in the bulk, have no effect on any weakly allowed cluster variables, or on their correlations. Along open boundaries, we find a particularly simple field structure; we calculate all $n$-point correlations, for any combinations of height variables and dissipative defect sites, and find that all heights and defects are represented by the same field operator.

19 pages, 10 figures. Error in field identification fixed. Referee suggestions incorporated, including new appendix. To be published in Phys. Rev. E

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• Scaling fields in the two-dimensional abelian sandpile model
• Modular transformation and boundary states in logarithmic conformal field theory
• Boundary height fields in the Abelian sandpile model
• Boundary States in c=-2 Logarithmic Conformal Field Theory
• Boundary states in boundary logarithmic CFT
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• Sandpile models in the large
• Three-leg correlations in the two component spanning tree on the upper half-plane
• Sandpile probabilities on triangular and hexagonal lattices
• Higher Order and boundary Scaling Fields in the Abelian Sandpile Model
• Multipoint correlators in the Abelian sandpile model
• Spatial Asymmetric Two dimensional Continuous Abelian Sandpile Model