Logarithmic two-point correlators in the Abelian sandpile model

Abstract

We present the detailed calculations of the asymptotics of two-site correlation functions for height variables in the two-dimensional Abelian sandpile model. By using combinatorial methods for the enumeration of spanning trees, we extend the well-known result for the correlation ${\sigma }_{1,1}\simeq 1/r^4$ of minimal heights $h_1=h_2=1$ to ${\sigma }_{1,h}=P_{1,h}-P_1{P}_h$ for height values $h=2,3,4$. These results confirm the dominant logarithmic behaviour ${\sigma }_{1,h}\simeq (c_h\log r+d_h)/r^4+\mathcal{O}(r^{-5})$ for large $r$, predicted by logarithmic conformal field theory based on field identifications obtained previously. We obtain, from our lattice calculations, the explicit values for the coefficients $c_h$ and $d_h$ (the latter are new).

28 pages

References in corpus (5)

• Associative-algebraic approach to logarithmic conformal field theories
• The logarithmic triplet theory with boundary
• Abelian Sandpile Model on the Honeycomb Lattice
• Pair correlations in sandpile model: a check of logarithmic conformal field theory
• Wind on the boundary for the Abelian sandpile model

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• Correlations in the $n\to 0$ limit of the dense O(n) loop model
• Watermelons on the half-plane