Deterministic Abelian Sandpile Models and Patterns

The Apollonian structure of integer superharmonic matrices

“…Ostojic [17] proposed classifying which periodic patterns occur in s n . Caracciolo, Paoletti, and Sportiello [4] and Paoletti [18] give an experimental protocol that recursively generates 2dimensional periodic "backgrounds" and 1-dimensional periodic "strings." While this protocol makes no explicit reference to Apollonian circle packings, we believe that the 2-dimensional backgrounds it generates are precisely the Laplacians ∆g C for C ∈ B.…”

Apollonian structure in the Abelian sandpile

“…Dhar [9] observed that the resulting stable configuration does not depend on the toppling order, which is the reason for terming the process "Abelian." When the initial configuration consists of a large number of chips at the origin, the final configuration has a curious fractal structure [3,11,[20][21][22] which (after rescaling) is insensitive to the number of chips. In 25 years of research (see [19] for a brief survey, and [10,25] for more detail) this fractal structure has resisted explanation or even a precise description.…”

Pattern Formation and Tropical Geometry

“…Line-shaped patterns, clearly recognizable in Figures 1, 2 along straight edges of the imaginable graph, explicitly came in the sight of researchers in Dhar et al [27] and Dhar and Sadhu [28] with an emphasis on a proportional growth phenomenon, and later, in Caracciolo et al [29] (see also [30,31]). These papers performed the analysis from the point of view of theoretical physics and explained the pictures based on experimental evidence.…”