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This paper reports a numerical study of how height variables in the two‑dimensional Abelian sandpile model are correlated at long distances. The authors compare their results to predictions from logarithmic conformal field theory (LCFT). They also extend the study to lattices where the usual analytical calculations are difficult, in particular the kagome lattice.
The Abelian sandpile model is a simple rule‑based model of grains piling and toppling that reaches a scale‑free, critical steady state on its own. Key observables are the height variables at sites and their bulk two‑point correlation functions, which measure how the height at one site is statistically related to the height at another site far away. In two dimensions the scaling limit of the model is expected to be described by LCFT, a version of conformal field theory that predicts power laws and additional logarithmic terms in some correlations because of nonlocal features of the model.
To get unbiased samples the authors avoid the usual Markov‑chain dynamics, which produces strongly correlated successive configurations and requires long burn‑in and thinning. Instead they use Wilson’s algorithm to generate uniform spanning trees efficiently and in parallel. Each spanning tree is mapped to a recurrent sandpile configuration by the Majumdar–Dhar burning bijection. This procedure produces independent samples and so eliminates sample autocorrelations, giving faster convergence of measured correlations. The authors also use fast Fourier transform based methods to evaluate correlation functions on large lattices.
On the square lattice and the honeycomb lattice the numerical correlations agree well with existing analytical predictions that come from LCFT and from graph‑theoretic calculations on spanning trees. For the kagome lattice the paper presents the first systematic numerical study of bulk correlation functions. For that lattice the authors also derive a closed‑form expression for the bulk probability that a site has height one, using the lattice Green function, and they use this expression as a cross‑check of the numerics.
This work matters because analytical derivations in the sandpile model depend heavily on specific lattice symmetries and on known Green functions. Numerical methods that give independent, well‑converged samples let researchers test universal features across different lattices and extract lattice‑dependent coefficients where theory is hard to apply. The approach should be useful for studying other two‑dimensional lattices and irregular geometries.
Important caveats are that the study is numerical and focused on bulk two‑point functions and a few lattices. LCFT predictions involve logarithmic terms that arise from the nonlocal mapping to spanning trees; comparing all such features requires careful analysis and, in many cases, lattice‑specific input. The kagome results are the first systematic numerical data for that lattice, and while the height‑one probability has an analytic check, a complete analytical treatment of all correlations on the kagome lattice remains more difficult.