Logarithmic two-point correlators in the Abelian sandpile model

TL;DR

This paper calculates two-point correlation functions in the 2D Abelian sandpile model, confirming logarithmic decay and providing explicit coefficients, thus linking combinatorial lattice results with conformal field theory predictions.

Contribution

It extends known correlation results to all heights and explicitly determines the coefficients for the logarithmic decay, bridging combinatorics and conformal field theory.

Findings

Confirmed logarithmic decay of correlations with explicit coefficients.

Extended correlation results to all height variables.

Provided new explicit values for coefficients in the asymptotic expansion.

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 $\sigma_{1,h} = P_{1,h}-P_1P_h$ for height values $h=2,3,4$. These results confirm the dominant logarithmic behaviour $\sigma_{1,h} \simeq (c_h\log r + d_h)/r^4 + {\cal O}(r^{-5})$ 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).