Analytic determination of non-square-lattice scaling-field coefficients

Determine analytically the lattice-dependent scaling-field coefficients \(\alpha_a\) and \(\beta_a\) in the logarithmic-conformal-field-theory decomposition of Abelian sandpile height variables on the honeycomb and kagome lattices, beyond the coefficients already known or numerically estimated.

Background

The paper represents the bulk height variables of the Abelian sandpile model as lattice-dependent linear combinations of a logarithmic field and its partner. Although the universal functional form of the correlation functions is expected to apply across sufficiently isotropic planar lattices, the coefficients in this decomposition depend on the lattice geometry.

Closed-form coefficients are available for the square lattice, whereas the honeycomb and kagome lattices have non-Bravais structures and more difficult Green-function and spanning-tree calculations. The paper extracts some coefficients numerically, but an analytic determination of the remaining coefficients would establish the lattice-dependent amplitudes independently of finite-size and fitting effects.

References

Can one determine the full $2$-primary Smith form of the same grids and locate the all-ones class inside that decomposition? Kuperberg's integral approach to matching matrices is a natural precedent . Spectral factorizations alone need not answer this question: equality between valuations of eigenvalues and local Smith invariants requires additional hypotheses , and even a Smith form must be accompanied by the coordinates of the chosen class. Maximum element orders in other graph families, such as strongly regular graphs, provide a comparison , but the exponent of a group need not be the order of its all-ones element. Could a monodromy-pairing calculation detect the latter directly, or explain the finer entrywise pattern in Figure~\ref{fig:n31-depth}?

— The 2-Adic Valuation of the Order of the All-Ones Class in the Sandpile Group of a Square  (2609.10625 - Akyar et al., 9 Sep 2026) in Section “Further directions,” paragraph “The full group and its distinguished class”

On the honeycomb and the kagome lattice the coefficients \alpha_a, \beta_a are not known analytically (apart from limited information on the height-1 sector); the universal form~eq:Cab_theory nevertheless applies, and the resulting lattice-dependent amplitudes are among the quantities extracted from our numerical data in Sections~\ref{sec:honeycomb} and~\ref{sec:kagome}.

— Universal correlations in the Abelian sandpile model  (2609.10352 - Liu et al., 9 Sep 2026) in Section 3.2, subsection “Analytical predictions: LCFT framework”