Multipoint correlators in the Abelian sandpile model

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.

Sandpile Models in the Large

“…Let us mention that the technique developed in [29] to enumerate the so-called cycle-rooted groves (which generalize spanning trees to spanning forests with marked points) currently provides by far the most efficient way to compute height probabilities, reducing the calculation of P 2 , P 3 to just a few lines (see Ref [31]). Most of the height correlators presented below have been computed using this technique.…”

“…The first few n-point correlators can be easily computed for arbitrary configurations of insertion points [31,33]. By construction, the 1-point function vanishes, σ 1 (i 1 ) 0 (the relation ( 9) is indeed the main motivation for the subtraction).…”

“…The mixed correlator of a height 1 and a height 2, 3, or 4 is harder. They have first been obtained in [34,35] by using classical graph theoretic techniques, and then reconsidered and extended in Ref [31] using the results of Ref [29]. Whatever the method used, one has to evaluate the fractionsX k (i 1 ) of spanning trees, as defined in Section 5.2, but on a lattice modified around the site i 2 where height 1 is located.…”

Sandpile probabilities on triangular and hexagonal lattices

“…where X G q (i) denotes the fraction of rooted spanning trees on G * such that i has q predecessors among its nearest neighbors (j is a predecessor of i on a rooted spanning tree if the unique path from j to the root s goes through i). A similar decomposition of spanning trees into subclasses can be utilized for multisite probabilities [16,28,33].…”

“…However, it would require the addition of O(p) extra edges to move the head of the zipper, and then evaluating the resulting sum for large p. Instead we found it more convenient to start from (3.7) and the integral representation (A.1) of the Green function. The calculation essentially follows the same steps as the corresponding one on the square lattice, made explicit in [33]. We therefore skip the details and give the final result in a case that is relevant to (A.7), namely G ′ (0,0),(p,2p+1) = G 0,0 − (0, 0) (1, 0) (1, 1) (0, 1) (0, 0) 0 − 2 3 G0,0+ 7 Table 5: Values of the Green function derivative of the triangular lattice around the origin, with respect to the zipper shown in panel (a) of Fig.…”