The Abelian sandpile model on the honeycomb lattice

Abstract

We check the universality properties of the two-dimensional Abelian sandpile model by computing some of its properties on the honeycomb lattice. Exact expressions for unit height correlation functions in presence of boundaries and for different boundary conditions are derived. Also, we study the statistics of the boundaries of avalanche waves by using the theory of SLE and suggest that these curves are conformally invariant and described by SLE 2 .

Sandpile probabilities on triangular and hexagonal lattices

“…A more appropriate way of writing ∆ to compute its inverse G is obtained by decomposing the lattice into unit cells [39]. Let a( r 1 , r 2 ) ≡ a( r 2 − r 1 ) be the 2 × 2 adjacency matrix for the vertices of the unit cells located at r 1 and r 2 , that is, Its inverse G depends only on the difference r ≡ r 2 − r 1 = (x, y), and is given [2] by…”

“…11). For both half-planes, which, following [2], we refer respectively to as principal and horizontal, we choose the reference site i = (0, p; A) with p ≫ 1. As for the full lattice, we select i, its three neighbors and the sink as nodes on a modified graph, here obtained by cutting the edge between i and its neighbor (−1, p; B).…”

“…The zipper extends up to infinity. y = 2/3, so the Green function reads [2] G cl (x 1 ,y 1 ;α 1 ),(x 2 ,y 2 ;A) = G (x 1 ,y 1 ;α 1 ),(x 2 ,y 2 ;A) + G (x 1 ,y 1 ;α 1 ),(x 2 −y 2 ,1−y 2 ;B) , G cl (x 1 ,y 1 ;α 1 ),(x 2 ,y 2 ;B) = G (x 1 ,y 1 ;α 1 ),(x 2 ,y 2 ;B) + G (x 1 ,y 1 ;α 1 ),(x 2 −y 2 +1,1−y 2 ;A) .…”

“…In particular, height probabilities and critical exponents have been investigated on such lattices in [10,21,27] using renormalization group transformations and numerical simulations. More recently, multisite height-one probabilities have been evaluated exactly on the hexagonal lattice [2]. All the results so far point toward a common set of critical exponents and scaling behavior for all lattices.…”

Height probabilities in the Abelian sandpile model on the generalized finite Bethe lattice

“…For example, to compute one-site probability P(x(v i ) = k), the first task is to characterize those spanning trees which correspond to recurrent configurations with x(v i ) = k, then the following work is to count those spanning trees. Using this idea, researchers have managed to obtain numerous exact results on some infinite lattices, such as the plane square lattice, [14][15][16] the Bethe lattice, 17 the triangular and honeycomb lattices, 18,19 the upper-half plane square lattice, [20][21][22] the d-dimensional hypercubic lattice. 23 Almost all of the above results are benefited from a technique developed by Priezzhev.…”