Multipoint correlators in the Abelian sandpile model

TL;DR

This paper advances the calculation of height correlations in the 2D Abelian sandpile model using a novel complex connection technique, providing more efficient derivations and new explicit correlator results consistent with conformal field theory.

Contribution

It introduces a complex connection method to compute height correlators in the sandpile model, extending known results and confirming conformal invariance properties.

Findings

Rederived 1- and 2-site correlators more efficiently

Computed new next-to-leading order terms in correlations

Extended correlator calculations to include arbitrary heights and boundary conditions

Abstract

We revisit the calculation of height correlations in the two-dimensional Abelian sandpile model by taking advantage of a technique developed recently by Kenyon and Wilson. The formalism requires to equip the usual graph Laplacian, ubiquitous in the context of cycle-rooted spanning forests, with a complex connection. In the case at hand, the connection is constant and localized along a semi-infinite defect line (zipper). In the appropriate limit of a trivial connection, it allows one to count spanning forests whose components contain prescribed sites, which are of direct relevance for height correlations in the sandpile model. Using this technique, we first rederive known 1- and 2-site lattice correlators on the plane and upper half-plane, more efficiently than what has been done so far. We also compute explicitly the (new) next-to-leading order in the distances ($r^{-4}$ for 1-site on the upper half-plane, $r^{-6}$ for 2-site on the plane). We extend these results by computing new correlators involving one arbitrary height and a few heights 1 on the plane and upper half-plane, for the open and closed boundary conditions. We examine our lattice results from the conformal point of view, and confirm the full consistency with the specific features currently conjectured to be present in the associated logarithmic conformal field theory.